Every experiment to date has determined that interial mass and gravitational mass are the same thing. Indeed, the equivalence of these two things is a bedrock foundation of general relativity.
Fundamental fermions each have one of twelve non-zero rest masses. Fundamental bosons each have one of four rest masses, with zero as one of the allowed values (belonging to photons, gluons and the hypothetical graviton, if there is one). We don't have any fundamental theory to explain the relationship of all of the fifteen non-zero rest masses of the Standard Model of Particle Physics to each other (we do have theoretical reasons for photons, gluons and gravitons to have zero rest masses), although we do have a formula that relates the mass of the W bosons to the mass of the Z boson, we know that there are some almost certainly non-random relationships between the fermion masses (such as Koide's formula for the charged lepton masses) although we aren't precisely sure who these numerical relationships arise, and there naiively appears to be a simple formula from which the Higgs boson mass can be derived from the W and Z boson masses (one half of two times the W boson mass plus the Z boson mass)that is a very close match to the tenatatively measured amount, although there is no consensus concerning why this relationship exists either.
It also seems to be the case, that there is an intimate relationship between the fundamental particle masses, the four parameters of the CKM matrix that governs the relative likelihood of particular flavor transitions via W bosons for quarks (including CP violating phases), and the four parameters of the PMNS matrix which codes the same relative likelihoods for leptons. The matrixes also seem to show some sort of relationship between the magnitude of the coupling constants for the three Standard Model forces (electromagnetism mediated by photons, the weak force mediated by W and Z bosons, and the strong force mediated by gluons) each of which is itself a function via equations and constants determined phenomenologically (rather than from first principles) of the energy level of the interaction in question which brings us back to the mystery of mass-energy all over again.
But, mass turns out to be a slippery thing. Mass is not simply additive in composite particles. Each of the couple hundred different possible hadrons has a very precise rest mass, but in composite particles bound by the nuclear strong force, the rest mass of the whole is generally not simply the sum of the rest masses of the component parts. Likewise, while total mass-energy in any system is conserved (with an E=mc^2 conversion factor), interactions via the nuclear weak force routinely do not conserve mass alone.
General relativity and special relativity add further complications. The relationship between mass and acceleration is a simple linear one at low velocities, but must be modified by a Lorentz transform at velocities approaching the speed of light. The relationship between mass and acceleration runs with a particle's kinetic energy levels.
Even more confounding, in general relativity, is the fact that forms of energy other than mass give rise to gravitational effects and are subject to the effects of gravity, even if they don't have any mass at all. A photon will follow the geodesic created by a gravitational field, even though it has no mass, and the flux of photons through a volume of space is part of the stress-energy tensor that gives rise to gravity in general relativity.
A mass field's linear momentum (in three dimensions) including its Lorentz boost factors, its angular momentum (in three dimensions), and the pressure it is experiencing (in three dimensions), in addition to its rest mass and the electromagnetic flux (more accurately four current) of energy in that volume of space also add to the stress-energy tensor.
The conventional stress-energy tensor of general relativity doesn't have terms for strong force flux and weak force flux, neither of which were known at the time it was formulated, but I don't think that anyone seriously doubts that fluxes of these forces contribute to the stress-energy tensor in the precisely the same way that fluxes of photons do.
Convention and personal preference dictates whether observed dark energy effects are modeled as a constant of integration in cosmological equations derived from the equations of general relativity, or as a real, uniform energy field that fills all of space-time and as energy which is a subset of mass-energy, gravitates. Physics already provides several fields which are present at nearly uniform levels throughout the universe - the physically observed and electromagnetic cosmic background radiation, the Higgs vacuum expectation value of the Higgs field, and the energy field implied by zero point energy (i.e. the amplitude in quantum mechanics for a particle-antiparticle pair to arise seemingly out of nothing in empty space), although none of these is a good match for the observed cosmological constant, or the observed overall flatness of space-time away from dense mass fields (as opposed to a strongly convex or concave structure of space time). Additional proposals are also out there, and as I understand the matter, the extent to which graviational fields (aka the background flux of gravitons in the universe) themselves, because they carry energy, give rise to gravitational effects isn't a question that I have seen a consensus answer to in the educated layman's and generalist physicist oriented literature (the question is subtle because "in general relativity the gravitational field alone has no well-defined stress-energy tensor, only the pseudotensor one.")
The standard description of the reason that efforts to describe gravity with a Standard Model plus graviton model is that the quantum mechanical equations of the graviton are not renormalizable, but given what I understand to be general relativity's BRST symmetry, (see, e.g. Castellana and Montani (2008)) it isn't obvious to me that this proposition is really true in a theoretical sense or in the sense that the equations actually break down in the UV limit, even if they may be impracticable to do calculations with by any non-numerical method we known outside special cases where simplifying ssumptions make an analytical solution possible. Castellana's abstract states (preprint here):
Quantization of systems with constraints can be carried out with several methods. In the Dirac formulation the classical generators of gauge transformations are required to annihilate physical quantum states to ensure their gauge invariance. Carrying on BRST symmetry it is possible to get a condition on physical states which, different from the Dirac method, requires them to be invariant under the BRST transformation. Employing this method for the action of general relativity expressed in terms of the spin connection and tetrad fields with path integral methods, we construct the generator of the BRST transformation associated with the underlying local Lorentz symmetry of the theory and write a physical state condition following from BRST invariance. This derivation is based on the general results on the dependence of the effective action used in path integrals and consequently of Green's functions on the gauge-fixing functionals used in the DeWitt–Faddeev–Popov method. The condition we gain differs from the one obtained within Ashtekar's canonical formulation, showing how we recover the latter only by a suitable choice of the gauge-fixing functionals. Finally we discuss how it should be possible to obtain all of the requested physical state conditions associated with all the underlying gauge symmetries of the classical theory using our approach.
(Abhay Ashtekar's reformulation of the equations of general relativity in the 1980s has been privotal to the field of quantum gravity.)
The concern that it might be necessary to retain background independence in an extension of the Standard Model with a graviton, see, e.g. here (although not necessarily discrete background independence, at least other than as part of a strategy to formulate the theory in a discrete setting and then use calculus to take the limit of that formulation as the minimal distance became infinitessimal) which is something that a naive quantization of a spin-2 particle on a Minkoski background modeled on other Standard Model quantizations can't capture that effect is a more serious concern. The fact that there is only a pseudotensor, rather than a stress-energy tensor for the gravitional field alone might also be a clue that general relativity's equation has a subtle defect in its formulation.
While the magnitude of Newtonian gravity is a function of rest mass only (and would imply a massless, color charge neutral, electromagnetic charge neutral, scalar spin-0 graviton), in general relativity, the overall magnitude of the effective gravitational force, as I understand it, is a function of total mass-energy in the volume of spacetime where it is being evaluated. Likewise, rather than being the simple radial attractive force of Newtonian gravity, in general relativity the direction in which gravity directs massive and massless particles alike, is modified from its radical attractive direction by a vector that incorporates the directionality of all of the particle motion, energy fluxes and pressure that are acting on volume of space-time in question.
The fact that both ordinary linear acceleration, and the acceleration induced by the force of gravity, which are identical in effect, also induces space and time dialation according to a Lorentz factor, further complicates the affair, which helps explain why the mathematics of general relativity is so challenging.
It has been hypothesized that the whole of general relativity and special relativity can be reproduced by simply quantum mechanical rules for a massless, electromagnetically neutral, color charge neutral spin-2 graviton (a tensor particle) that couples to everything with mass or energy, and the spin-0, CP-even, 125 GeV +/- 2 GeV, electromagnetically neutral, color charge neutral Higgs boson (a scalar particle), although to my knowledge, no one has ever successfully proposed an operational realization of this hypothesis that has been rigorously shown to be equivalent to the equations of general relativity or some variant of those equations that is empirically indistinguishable through some slight technical tweak to the theory (such as Einstein–Cartan theory which adds torsion to the metric which allows gravity to respond to spin angular momentum in a way that the original formulation does not, or the Brans–Dicke theory of gravitation, which is a scalar-tensor theory and hence naively more directly parallel to a Higgs boson-graviton formulation in quantum mechanics).
Modified gravity theories attempting to explain dark matter effects which are consistent with general relativity in all domains where dark matter effects are negligable, are generically scalar-vector-tensor theories (the Bekenstein direct derivation of Milgrom's theory is dubbed TVS, while many versions of Moffat's theory that attempts to do something very similar in a slightly different way, prefers the order SVT), and were these theories to be quantitized, would presumably require, in addition to a spin-2 graviton, a spin-1 gravitovector (presumably massless, color charge neutral, and electromagnetically neutral), and perhaps also a massless spin-0 scalar graviton if the Higgs field couldn't be appropriated for that purpose.
Loop quantum gravity proposes a discrete space-time structure from which the four dimensionality of space-time and locality are merely emergent properties that are ill defined at the quantum level. Rigorous, but theory dependent tests have the discreteness of space-time have so far demonstrated a continous space-time structure at scales that would appear to be well below the Planck scale below which many direct measurements of distance and time associated with particles becomes inherently uncertain. Quantum mechanics exhibits a phenomena called entanglement which fit some definitions of non-locality, although entangled particles must share of speed of light space-time cone from a common point of origin in space-time bound and there are theoretical questions over what this bounded form of non-locality means and what can be achieved with it in terms of information transfer.
LQG tends to envision mass as someting sort of like clumping of nodes of adjacent points in space-time together. Some versions of it have a graviton that emerges from the equations and propogages.
Supersymmetry models, like the Standard Model, does not include gravity and are formulated in Minkowski space. The gravitational extention of supersymmetry models is generally called supergravity (SUGRA) and string theory/M-theory generally attempts to embed supergravity theories within its overarching substructure and naturally predicts the existence of a spin-2 particle associated with a graviton. String theory uses extra-dimensions, in which gravity interacts more easily than the four observable dimensions, as a mechanism by which to turn a force which is much weaker than the other three fundamental forces in the context of systems with small numbers of particles interacting with each other into just another manifestation of the an underlying fundamental force whose symmetries are broken by branes, dimensional compactifcation and other mechanisms that are not always well defined.
Neither general relativity nor special relativity nor Newtonian mechanics and gravity, contemplate a physical, aether-like Higgs field that gives rise to interia. Newtonian mechanics employs the low velocity limit of special relativity relating force and acceleration of F=ma as a low of motion rather than a substance, and takes the fact that matter has mass in amounts to be empirically determined as axiomatic. The equivalence of gravitational mass to inertial mass, and of gravitationally induced accelerations to other accelerations is a core axiom from which that theory is derived, and while general relativity does conceptualize mass as a sort of crystalized energy that factors into the Lorentz equations in a manner different than energy not in the form of mass does, general relativity does not address the question of what process causes energy to crystalize into mass. None of the classical theories of gravity and mechanics has an aether-like field that gives rise to inertia like the Higgs field.
The Standard Model is formulates in Minkowski space, where special relativity applies, but there is no gravity and no curvature of space-time that flows from gravity, although ad hoc applications of classical general relativity in a non-systemic way to the equations of general relativity in circumstances where general relativity effects are intense, for example to understand Hawking radiation from black holes, has been attempted with success. Among other problems with this approach, wave-like field theories do not naturally transform into particle-like theories in curved spacetime, and the acceleration of the observer influences the observed temperature of the vacuum.
It also is worth pointing out that despite the new development of the Higgs boson, parallels between the QCD equations and gravity seem strong than parallels between electroweak equations and gravity, even though the electroweak equations seem to be what is imparting rest mass to the fundamental particles in the Standard Model. The QCD connection is particularly notable given that 99% of baryonic mass arises from gluon exchange in hadrons. One could imagine, for example, a quantum gravity Lagrangian equation that was somehow related to the square of the QCD Lagrangian plus the square of the electroweak Lagrangian, weighted relative to the contribution of each set of equations to the source of the gravitational mass. A 1% contribution to gravity from electroweak sources which frequently were proportional to QCD sources, since weak force decay at any given moment in low energy systems isn't much of a flux and the proportion of particular kinds of fundamental particles (up and down quarks, electrons, neutrinos and unstable fundamental prticles) ought to be relatively uniform everywhere, might make that component of a true law of gravity invisible.