I've never owned an AI and my time behind a number of their offerings is less than 200 rounds total, but I am on my second DTA. I think the DTA is a great rifle, but I can also appreciate what it does well as well as the shortcomings it has.
Before you buy a DT, I would highly suggest you spend some time behind one and really manipulate the bolt, swap a mag, etc. In my experience, people either love the position of the bolt or they hate it. There is really no in-between and if you don't immediately say, "Yeah, this isn't bad! I can get used to this!" to yourself, chances are you never will.
What I love about my DTA: It's incredibly accurate. Zero shifts between barrels are repeatable. Just yesterday, I swapped from my 6.5 Creedmoor to my 308 barrel. I dialed my adjusted DOPE and had bullets on steel. Barrel swaps are pretty quick. If I'm not changing the bolt face, barrel swaps can take right around a minute. If you need to swap the bolt face, add another minute or so. Your choice of calibers is almost limitless, though there are some that won't feed or eject correctly. The SRS or Covert can go from 223 to 338 with just a barrel and bolt face swap. If you want 50, any of the CheyTac rounds, etc, you'll have to go with the HTI, just like you'd have to go with the AX50 for the AI counterpart. The trigger is very good, borderline great considering it's a bullpup, and adjustable for trigger shoe position, creep, and pull weight. It's incredibly compact and very well balanced for most of the caliber/barrel length choices. Like others have said, if you're hunting with it, it's so compactyou can throw it in a backpack, especially if you go with the Covert and a shorter barrel, and hike to your shooting position.
What I don't care for: Since StrykerVet mentioned it, I'll start with the magazines. HTI mags are expensive. Hell, even the short or long action mags are expensive, especially for what you get. 6-round SA or LA mags are $90, 8-round LA and 10-round SA mags are $140, and the 5-round HTI mags are $255 (about $100 more than the AX50 mags), and no one but DT makes them, so it's not like you can find a less expensive alternative from Alpha or someone else like you can AI mags. Fit and finish on the DT is good, but not great. I may be off base here, but I would attribute this to AI having a 30 year head start on DT. I would hope that DT will continue to improve as the years go by, and they've shown they'll update their rifles, but for now the AI just exudes quality, while the DT is still a bit rough around the edges. Barrels for the DT can be expensive. Some of the top smiths are charging $1300+ for a DT barrel. I go with a guy on the Hide that is very popular with DT shooters who does top notch work and I'm still paying around $1k for a barrel. Want the 223 conversion kit from SAC? That's $1700 for the barrel, 223 bolt face, and magazine. While you can shoot most cartridges in a DT, you can't shoot every cartridges. I'd love to be able to shoot a 6 Dasher in my DT, but it won't eject correctly 30% to 40%. I know there are a few other rounds that just won't work with the DT.
Overall, I'm happy with my DT. I admit I lust after the AXMC, but even if I sold my DT, barrels, bolt faces, mags, etc, and received top dollar, I'd still need to come up with another $2k to $3k to buy an AX with all the additional barrels and accessories. To me, the price differential is large enough to keep me shooting my DT.