Amazingly, Illustrator allows you to output to multiple formats. However, it's vector based work allows your designs to work in far wider range of places beyond just the pixelated world of the web.

Logos, icons, or other illustrations should be done in Illustrator. If you do a 500px logo in Photoshop, you are effective fixed at 500px.

If you're merely referring to using Photoshop as a glorified layout editor for creating a quick rendition of the site, I guess that's fine. But then it's overkill by a large margin for the task.

If you expect to do graphics for the web, learn illustrator. If you except to merely mock up a site to later have it used merely as a reference for CSS/HTML, then their is less expensive software out there.