The concepts of vacuum energy aka zero point energy, dark energy, the Higgs vacuum expectation value, and the scalar inflation field of cosmology, are in principle distinct, but because all are conceptualized as properties of what is otherwise a vacuum, and the concepts are naively inconsistent with each other, the relationship between them is cryptic.
One of the unsolved problems of physics is why the "vacuum expectation value" of the Higgs field is so many orders of magnitude larger than the average density of dark energy implied by the cosmological constant, which seeks to reconcile a couple of these concepts.
Dark energy is, however, of the same order of magnitude as dark matter and of ordinary matter, based on the lamda CDM model of cosmology. This is vastly less than the amount that would be expected if there was a Higgs field vev of 246 GeV throughout the entire universe.
The answer may be one of cause and effect. It is common sense to think about quantum fields as being created by bosons emitted from fermions.
Perhaps we should think about the Higgs field as something that is created by the Standard Model fermions, just as we think about the electromagnetic field as created by quarks and charged leptons, and strong force QCD fields as created by quarks, and gravity is created by matter-energy particles in empty space (apart from the cosmological constant).
Dark energy seems like it is everywhere. But, a model in which dark energy is present merely in the vicinity of Standard Model fermions (massive W and Z bosons must always be near the Standard Model fermions that emit them and absorb them because they are so short lived), would be virtually indistinguishable.
Because at the relevant scales at which we observe dark energy effects, the universe is essentially topologically flat and homogeneous. Dark energy that was clumped around fermionic matter wouldn't look different from dark energy distributed uniformly at a rate of the expansion of the universe scale, but the fermionic matter distribution itself is quite smooth.
If Standard Model fermions generate the Higgs field, then it also makes perfect sense that the aggregate energy of the Higgs field should be on the same order of magnitude as the aggregate mass of the fermions in the universe. Higgs fields would only exist in the vicinity of these particles. The notion is similar in concept, although quite distinct from Mach's conception of gravity.
Phenomenologically, this would also suggest that empty space really and truly is empty, that vacuum energy is a flawed oversimplification that doesn't apply to truly empty space, and that the universe is particles all of the way down. But, another way, while the Higgs vev in the vicinity of Standard Model fermions is 246 GeV because the Standard Model fermions generate this Higgs vev, in deep space far from any fermions, the Higgs vev is asymptotically 0 GeV has the distance from any fermionic matter grows great enough.
Also, if there was a high energy Higgs field in the vacuum, why wouldn't it give rise to Higgs bosons that would decay into fermions making that vicinity no longer a vacuum?
Footnote: The relationship between resonance width and mean lifetime is t=1/W. The W boson, and Z boson have resonance widths of about 2.1 GeV and 2.5 GeV respectively corresponding to mean lifetimes on the order of 3 * 10^-25 seconds) which gives the weak force an effective range on the order of the size of an atom (the top quark has a mean lifetime of about 5 * 10^-25 seconds and a slightly smaller resonance width). The Standard Higgs boson resonance width is on the order of a MeV. Thus, the mean lifetime of a Higgs boson is about 1000-2000 times that of a W or Z boson (about 10^-22 seconds, give or take), and naively a range for its interactions were it emitted on a similar basis on the order of 1500 times the size of an atom.