Axe FX is suppposedly a pretty deep editing experience, as far as virtual amp modelling goes; though I have never actually gotten my hands on one. One of my buddies plays guitar in the band "Digital Tape Machine", and he's been working with the Axe FX, and seems happy.
While I don't know about the Axe-FX, I can tell you my personal experience with the DT25+HD500, and more recently with the JTV.
The ONE thing that the DT+HD500 does, that *nothing* else out there can do, comes from the Bogner portion of the DT design. When the HD and the DT are 'L6-Linked' together, you enable analog and power amp related switching to occur, so instead of just modelling the characteristics and response of the amp models, you are reconfiguring actual relays, switches, and various actual components on the fly.
As far as quality, it's hard to say - I would guess that for the price of an Axe-FX, you get what you pay for! Though, it's also still 100% 'virtual reality'.
The HD500, as far as amp and fx models go, is quite good. I have used the HD500 since 2010, and only recently upgraded to a DT25 1x12 combo. I didn't want the weight of a DT50, and the volume / power was a bit more than I needed, so the DT25 was a good option for me.
Thus far (I just added the DT25 and the JTV59 beginning of 2014) I am quite happy with what I am hearing, right out of the box. I used to endlessly tweak patches to get a decent sound, and now I am able to spend alot more time playing, and a lot less time patch editing.
In terms of the Axe-FX, my primary concern for myself, beyond the price tag, is the literally unlimited options. The Ax-FX can do things, of course, that the POD series will never likely delve into. For me, I think I would spend all my time learning to interface with the gear; though that can be exciting, it can prevent one from focusing on the actual music.
I was looking at the ElevenRack recently, since I was shopping ProTools, and they are bundling those two together for a decent price lately; essentially it's ony $100 more than the cost of the software to add the 11R. Not sure where they sit in the quality range; it's in the same basic realm, of amp and FX modelling as the HD and the AxeFX; 24bit, 96k, etc.
If you get a chance, the key to truly experiencing whether you will like the Bogner/L6 DT + HD, is to make sure you are using the L6 link. It does require a difference in understanding of what you are doing to your sound. I have had to re-do all my patch building ideas from the ground up.
The preamp portions of the HD amp models are all that is used when L6linking, and building patches using the 'pre' choice of the models. The 'full' version include power amp modelling, that is redundant when utilizing the actual power amp functions, which is where the Bogner portion of this rig really shines.
While the HD500 does a great job of direct amp, cabinet and mic modelling, I can say I just prefer the tonal warmth of the manner in which the HD and the DT share these duties. I realize this isn't as specific as you might want, so hopefully some of the others here who have had their rigs for longer than I can offer more specific insight.
The other bonus you get with the HD+DT rig using the L6Link, is the ability to chain up to four of the DT amps, and configure on a per-patch basis, what each amp will receive. You can set them as a basic stereo pair, left right. OR you can use dual amp configurations, sending each DT a different amp model. Again, the HD is only modelling the preamp portions of these amps, when choosing the PRE models for use with the DT. Tone stacks, gain, color, shaping. All the good stuff - pentode, triode, class A, class A/B, that is handled by the Bogner.
I guess once you factor in the cost of an HD+DT vs the cost of an Axe-FX + power amp, you are similar in price point.
Also, the newer StageSource speakers are also able to handle the L6Link features of the HD and DT rigs, so you could branch out even furthur into connection possibilities, by linking the HD to a DT and to an L2T or the like.
Lastly, if you get a chance to L6Link an HD to a DT, do yourself a favor and try out a JTV as well. I can honestly say my JTV59 is a revelation in guitar flexibility, and for being a Korean made axe, is really quite nice. For pure simplicity of being able to change a whole bunch of things with one foot switch on the HD, it's a nice option - you can even store the choice of magnetic pickup or modelling pickup for the JTV within the HD500 patch. Meaning, switches the DT to a Fender style amp while changing your JTV to a strat. Next patch changes the DT to a Marshall Plexi and changes your JTV to the onboard humbuckers, or to a Les Paul model, etc.
I upgraded to the HD500 from an X3Live, so I know how you feel, in terms of hearing the new HD amp modelling as compared to the previous offerings.