Security - image files can contain something other than an image, and hackers can attempt to (ab)use an upload script to upload code which can be run by the server, thus gaining access (there is a recent vulnerability in IIS [webmasterworld.com] that demonstrates this). Make sure you are checking the contents of the file and that it will not be executable.
One advantage of uploaded avatars is that you have more control over what displays. If the avatar is hosted remotely in a place controlled by the user, it's easy to change an acceptable image to something problematic.
A more complex alternative would be to allow uploading to another location you control and linking from the forum to those images. This keeps your prime location secure and also lets you offload some bandwidth if that's an issue.
I have avatars invisible to guest traffic. I also have them set to be uploaded to my server only. There are a few reasons for this, but it's mainly to prevent the kind of attacks that can be triggered from hotlinked images (damned if you do, damned if you don't) and another key issue is that offsite avatars can slow down your pages.