
The one on the left does not have anti-aliasing, the one one the right does.
Anti-aliasing kind of blurs edges to make them softer.
Each pixel of a picture usually has three values: Red, Green, Blue, and Opacity (alpha) when it is supported. GIFs use a palette (index) to reduce file size. A GIF pic has a maximum of 255 different colors and uses them as in a number painting (dunno if that's how it's called in English; the kind of paintings with outlines of zones to fill in with a set color). GIFs also only support a 1-bit Alpha channel, which means the fourth value of every pixel is either 0 or 1 (invisible or opaque).
By comparison, PNGs can have more than 16 millions of colors and a full 8-bit alpha channel (256 possible values of opacity for every pixel).
This means GIFs can't have anti-aliasing in the alpha channel, but PNGs can.
That's also why when you edit GIFs, you'll want to make them RGB (unindexed) beforehand so you can use whatever color you want and not only the ones in the existing palette.
Hope this helps.