I'm comparing caves between 0.43.05 and 0.44.02, and there is one significant difference: they're a lot larger.
The old caves are 1*1 mid level tile "boulders" on the pre embark map (assuming the world gen setting was to display them). This is also reflected in the cave feature (buried deep down in the df.global.world.world_data.feature_map structure), where the start x = end x and start y = end y.
The new caves are 3*3 mid level tiles with a "boulder" at the center and a "wall" around it (as displayed by the Phoebus tile set). This is also reflected in the cave feature, where end x = start x + 2 and end y = start y + 2. I suspect that also answers the question of whether the Kobold cave remake affects regular caves: it probably does (someone would have to send an adventurer to one or embark on top of one to check).
Apart from the size change, DFHack does not show any change to the cave feature intself, but that doesn't prove there isn't any: it''s possible potential additions haven't mapped/examined (and neither of the caves I looked at were inhabited by Kobolds).
Edit: The cave feature of a cave where Legends Viewer claims Kobolds live looks the same as the non Kobold cave looked at earlier.
Edit 2: Embarked on top of a non Kobold cave, and yes, it's definitely different. This cave had above surrounding ground open "rooms" in a mound, and the path down to the cavern is more "roomey" than the narrow path of previous versions, although I wouldn't exactly call these wider features rooms.