Figured I'd chime in with my experience. I've been playing electric for about 20 years now, and tube amps for just as long (started with a used Peavey Ultra 212 I got for like $300 on eBay around 2002 -- what a beast!). I've had most of the modelers starting with the Line 6 POD, which was cool for what it was, but compared to the stuff available for the same price today, is pretty awful. Also owned various POD iterations through the years, Vox's modeler, and the el-cheapo Behringer POD knockoff that actually sounded better than the POD if my ears remember correctly.
I currently run an Axe FX 3, primarily for silent / studio monitor recording since I moved to an apartment. Before I moved, though, I had a whole basement to myself, and ran it through my Fryette PS2 for a tube power section, with the power-amp modeling turned on.
When the Axe 3 first came out I would say it wasn't quite "there" as far as making me feel like I was playing a tube amp. It was great, and sounded great, but the feel came into the equation with the many firmware updates Cliff released up until today. Now, I swap between my tube heads and the Axe depending on my mood (the heads I own are amps not modeled in the Axe), and I honestly wouldn't be able to tell you the difference. The amps in the Axe sound and
feel like tube amps. And they have the wonderful advantage of not being prone to noise like tube amps are, which makes recording so much friendlier.
What one needs to realize is that every tube guitar amp, even with the same circuit and components, will be a little different sounding at the same settings. Just like guitars. Even the mains electricity can make an impact. I just today did some troubleshooting with my Friedman BE-50 that was sounding a little off and lower gain than expected. It prompted me to unpack my Furman voltage regulator and pop it into my rack, and what do you know, it sounds much more like the amp that I remember now that it's getting a regulated 120V. It's still a little lower gain than the BE-100 models in the Axe, but from all the YT clips I've listened too, I'm starting to think that is normal (Dave Friedman, of course, being the awesome guy he is, offered to dial my amp in if I sent it to him).
What the Axe gives me is utter consistency. I can always count on the amp models all sounding the same from day to day, whereas with tube amps, they can be finicky depending on the circuit (especially Marshall style circuits, in my experience). And when I was able to run it through my Friedman 4x12, it was glorious. I certainly couldn't tell the difference. The key with that kind of setup is knowing the parameters in the Axe and how to set them to not overhype the bass and treble frequencies, as a tube power amp will recreate that naturally, whereas the Axe seeks to model that kind of amp <-> speaker interaction by simulating the different impedance loads different cabs will place on a tube amp.
My patches these days are set up with an input for my loaded amps, like my BE-50, KSR Gemini, and Landry LS100G3, as well as an input to the Axe itself. That way I can keep the patch the same for FX, like the cab IR and reverb, delay, etc, while switching pretty seamlessly between my tube heads and the Axe itself. It's a great setup. That said, if I didn't have such a problem with GAS and had to choose one thing, I would choose the Axe since it gives me the most versatility and bang for my buck.
And with all that said, I still do recording with a miced cab since my rack setup is a PITA to move given its weight. I currently have a Mesa Fillmore 25 miced up at my office that I use for video recordings. At this point, it's the tool for the job. Use what works for you, etc etc etc