It's not so much the sound per se, a lot of has to do with more accurate tuning.
It's always been my opinion that, compared to standard hoops, 2.3 hoops offer more consistent tuning, the drum stays in tune longer and doesn't de-tune as frequently, and provides more clarity on rim shots and cross stick "clicks".
They provide more accurate tuning because of their rigidity. Because it's stiffer, it's going to sit on the head more evenly, rather than being easier to flex from an uneven tension spot. It will also help each tension rod hold it's tension because of that. Also, when mounted and ready to be played, 2.3 hoops come down a little further on the shell, pretty much even with the "rim" on the head, which makes it more "one" with the head on the shell, rather than 1.6 hoops, because on those the "rim" of the head sticks out beyond the bottom of the hoop.
They also make the drum heavier. Comparing two identical drums with one having 1.6 hoops and the other having 2.3 hoops, the drum with 2.3 hoops will be heavier and feel beefier and "better built".
Having said that, I use 2.3 hoops on the top and bottoms of my toms and snares. I don't really notice any difference in sound on toms, except that sound a little more "pointed" and beefy". On snares, the rim shots and cross sticks are much more pronounced and the whole drum just sings more. I used to think it wouldn't be necessary on the snare (bottom) side, but it comes in handy when sitting in a snare basket and you tighten up the basket. I know you're not supposed to use a lot of tension on the snare basket (and I don't), but it seems like the basket grips better and with less tension, plus the rigidity of the 2.3 helps hold the tension in place, even when light pressure from the basket is applied to it.
The biggest thing to remember about 2.3 hoops is that they will do the things mentioned above, but won't make any significant changes in the sound of your drums by themselves. Head selection and head tension are still 95% of what makes your drums sound like they are drums. In my opinion, 2.3 hoops are always a good upgrade. I don't see any drawbacks in using them at all.