I currently have a RPi 2 which has KODI installed and I use purely as a media center. I've noticed a few video's seem to stutter on it though; some 720p content stutters, and most 1080p content stutters.
Given the RPi 3 uses exactly the same GPU, is the RPi 3 any better, or exactly the same?
the pi 3 does not use the same CPU as the original pi 2. they revised the pi 2 to use the same cpu as the pi 3 to reduce production costs after the 3 was released, but the pi 2 version is clocked slower. the pi 2 runs at 900MHz whereas the pi 3 runs at 1200MHz, so there is some extra performance available.
However if you haven't already looked into it, buying the license codecs for your pi will help, at the bottom of this page
Note that the license will only work for that specific pi, so if you choose to upgrade to a 3 later you would need to buy new codec licenses if you wanted them.
the licenses allow the pi to use the GPU for decoding the video, instead of the CPU for decoding and the GPU for pushing pixels (as it comes out of the box)
omxplayer, which I think works without a licence...