The problem is how GIFs render transparency. What might help
is to change the "matte" color found in the Optimize panel. It's
probably set to white. Set it to match the background color of
wherever the GIF is going to go.
Because GIFs can't have semi-transparency, then anti-aliased
(smoothed) edges basically have to have a background color baked
into those pixels, that's what the matte color it. However, if you
use PNG32 or PNG8 with Alpha Transparency, it will remain smooth.
The only problem is the same problem with most things on the web:
Internet Explorer. Alpha Transparency doesn't work in Internet
Explorer 6 or below, without using some crazy hacks you'd have to