A quantum gravity theory based upon a massless spin-2 graviton should, in the classical limit, reproduce general relativity (GR) (I have yet to see any really rigorous proof of this piece of folk wisdom). But, such a theory isn't and can't be, completely identical to GR, although devising an experimental test of whether it is one or the other is a question that has stumped physicists so far.
There are some pretty generic qualitative differences between classical GR and any theory of gravity based upon graviton exchange. Here are fifteen of them. In a quantum gravity theory:
1. Gravitational energy is localized (this is not true in GR).
2. Gravitational energy is perfectly conserved (this is not true in most interpretations of GR).
3. Graviton self-interactions and graviton interactions with other particles would look the same mathematically, while in GR gravitational field self-interactions do have an impact on space-time curvature, but while all other kinds of mass-energy inputs make their way into Einstein's equations via the stress-energy tensor, gravitational field self-interactions are treated differently mathematically.
4. Gravitons deliver gravity in tiny lumps, while space-time curvature does so continuously; i.e. sometimes graviton should act like particles instead of waves, while GR has only wave-like gravitational behavior.
5. Gravitons ought to be able to exhibit tunneling behavior that doesn't exist in classical GR.
6. A graviton based theory is stochastic; GR is deterministic.
7. It is much less "natural" to include the cosmological constant in a graviton theory than in GR where it is an integration constant. In a quantum gravity theory there is a tendency to decouple dark energy from other gravitational phenomena.
8. In a quantum gravity theory, gravitons couple to everything so a creation operator from a pair of high energy gravitons could give rise to almost anything (in contrast, photoproduction can give rise only to pairs of charged particles that couple to photons); likewise any two particles with opposite quantum numbers could annihilate into gravitons instead of, for example, photons. Neither creation nor annihilation operations exist in GR in quite the same way, although seemingly massive systems can be converted into high energy gravitational waves.
9. In some graviton based theories, properties of a graviton must be renormalized with energy scale like all of the SM physical constants; in others there is a cancellation or symmetry of some kind (probably a unique one) that prevents this from happening. One or the other possibility is true but we don't know which one. GR doesn't renormalize.
10. In graviton based theories lots of practical calculations require approximating infinite series that we don't know how to manage mathematically; in GR, in contrast, infinite series expressions are very uncommon and the calculations are merely wickedly difficult rather than basically impossible.
11. In GR singularities like black holes can be absolute; in a quantum gravity theory they can be only nearly "perfect" but will always leak a little, because they are discontinuous and stochastic.
12. In quantum gravity it ought to be possible to have gravitons that are entangled with each other, while in GR this doesn't happen.
13. In quantum gravity with gravitons, the paradigmatic approach is to look at the propagators of point particles; GR is conventionally formulated in a hydrodynamic form that encompasses a vast number of individual particles (although it is possible to formulate GR differently while retaining its classical character).
14. In quantum gravity, calculations for almost every other interaction of every kind need to be tweaked by considering graviton loops; in GR the gravitational sector and the fundamental particles of the Standard Model operate in separate domains. For example, even if Newton's constant does not run with energy scale due to some symmetry in a quantum gravity theory, the running for the strong force coupling constant with energy scale would be slightly different than in the SM without gravitons.
15. Adding a graviton to the mix of particles in a TOE qualitatively changes what groups can include all fundamental particles that exist and none that do not; while in GR where gravity is not fundamental particle based, it does not.