In addition the the photos provided by PretzelNymph, here are a series of photos I took, showing a side-by-side comparision of the Tamagotchi nano and Chibi Tamagotchi (Japanese version of a Tamagotchi mini.)
They're about the same height & width, but the nano is much thicker. The nano has a higher resolution display.
In the above photo, the screens appear to be the same size, but actually, the nano's screen is a bit bigger than the chibi's. Their relative screen sizes can be judged in the top-most photo.
I think some of the reason for the nano's increased thickness is that it takes 2 LR44 batteries, which are thicker than the CR2032 used in the Chibi.
Also, as hinted at in the 3rd and 4th photos, the rounder design sets the screen lens in a bit, so hopefully it'll be less likely to get scratched.