There seems to be little (or no) pixelation at the edges of your web images, but plenty of JPEG muddiness.
That is rather strange for a GIF.
Are you sure the image size is the same when you upload it? A raster image at a wrong size in pixels x pixels is a known source of pixelation.
To avoid any issues at the edges in connexion with a micmatch between Matte colour and (possibly varying) background colour, you may use PNG instead of GIF, if that is allowed.
Inside Illustrator the screen is always redrawn at the current resolution and therefore is always crisp.
That can't happen anymore as sonn as you export to a rasterbased format such as GIF or PNG or JPEG. The image is then frozen at the given resolution and therefore of course pixelated.
Aside from trying some dithering in the Save As > GIF dialog, It looks pretty good on my screen. Are you viewing the file on a "retina display" by chance? Is Antialiasing selcted in the original Illustrator file? Monika is right, the file does become rasterized at 72ppi when the GIF is produced. Are you seeing really obvious stair-stepping jaggies in the browser? Is the logo being sized up in the browser?
I did try saving it as PNG, BMP, TIF, GIF, and JPEG and I keep getting the same pixelate result. I do not believe that it is resizing the image. I did try it at several different sizes to check but it still looked the same no matter what size I made it. I know that there are some issues with pixelation on facebook but I didn't know that this would be the case on other sites as well. Could it be because the color is so saturated? Which just makes it more noticeable maybe.