One of the problems with PNG is that many applications default to full-color-with-alpha-channel as the only save method, even though PNGs can do paletted images just as well as GIF and end up basically the same size. So people look at their PNG files and wonder why they are so big compared to the old GIFs because their application sucks.
Edit: I just tried it with your example:
% giftopnm favicon.gif | pnmtopng > favicon.png
pnmtopng: 5 colors found
% ls -l
-rw-r--r-- 1 jandrese jandrese 216 Aug 7 17:33 favicon.gif
-rw-r--r-- 1 jandrese jandrese 255 Aug 7 17:33 favicon.png