i have yet to try out the HDs or an axe fx (i really want an axe fx), but i'm a long time POD X3 user... as far as i can tell, the HD has improved in some areas, and kinda fell back in others...
the main advantage i can see is that the HD has anything-to-anything routing (somewhat... not as good as what the axe-fx does), whereas the X3 has few effect placement options...
also, they've seperated the preamp models from the poweramp models, which is a good thing for anybody using it with a real tube amp... on the other hand, there are far less amp models... the X3 has over 70 and i think the HD only has 16, but that might change as updates/model packs start to become available...
the thing the axe-fx has over just about any processor is that it's not a "modelling" processor (although some of the amp sounds are based on actual amps)... like with the POD, you have 3 or 4 different choruses that are modelled on real chorus pedals... with the axe-fx, there is only one chorus, but it has so many parameters to tweak that you could get the sound of those same 3 or 4 choruses that the POD has, plus just about any other chorus, or invent your own sound... plus i suspect the converters and other internal electronics are better than anything out there... they're charging an arm and a leg so it better have better converters...