I can offer some good insight here. I looked into this very question on the way toward purchasing an L-00 Studio, as I am extremely picky about neck feel and NOT a fan of low-profile necks.
Although there are subtle variations from neck to neck, in general, the Advanced Response neck is indeed noticeably fuller in feel than the traditional Slim Taper neck.
The AR necks tend to range in depth from around .86" to .90" at the first fret. The vast majority that I've seen spec'd have been in the .87"-.88" range at 1st fret. My L-00 Studio measures about .877" at the first fret. A nice full feel. An AR neck on one particular Hummingbird Studio I tried was unquestionably fat. That one must have measured around .90". Almost a Banner feel to it.
In contrast, typical Slim Taper necks tend to run around .80" to .84" at the first fret (i.e., instruments such as the J45 and L-00 Standards). Closer to the '60s Gibson profile.
One thing both neck styles have in common is that they don't get very obviously thicker up through the 9th fret. In that sense, they both have a "slim taper" quality. However, the general feel of the AR necks, and indeed my L-00 Studio as an example, is much fuller than any Slim Taper neck I've ever handled.
Also worth mentioning is that the AR neck has a subtle V feel at the first fret that gradually flattens to a C around the 9th fret. The radius is 16" as mentioned earlier.
All of this considered, the Advanced Response neck, to me, feels like a hybrid of the late-'50s Gibson rounded profile and the '70s Martin full profile soft V. It is not a huge 1" bat neck but it definitely feels more chunky and full to me than your typical modern low-profile neck...and, for me, that's a really good thing. Gibsons from the late '80s and early '90s often have a similar feel--a fuller C profile. I have a '91 J30 with that profile. It seems to have started to thin out again around the mid-'90s.
Two great resources for gauging neck depths are the websites for Wildwood Guitars and The Music Zoo. Both sites routinely post neck-depth specs on instruments. Chicago Music Exchange also posts neck depths on used and vintage instruments. All of these will display commonalities among neck profiles across different brands, models and years. It's a great research tool if you are--again, like me--very picky about neck feel and more a fan of fuller-feeling necks.
Hope that helps.
-Mike