>They don't include animations, they don't accept user input, they don't connect to servers and stream files and send updates
A PDF file is much more than just words and pictures. There are forms which accept user input, weird embedded content, and other strange things. Although I am not familiar with the spec, I have seen the features that Adobe Acrobat lets you do with PDFs and it seems pretty monstrous.
To clarify, there are several security issues that arise from all of this , and as others on this page have pointed out, that is a good reason to make use of the existing infrastructure around browser sandboxing.
 - http://duckduckgo.com/?q=adobe+pdf+vulnerabilities