Forums allow for a more in depth thoughtful responses. In a chat room you can't measure your words, research answers, format test include graphics or edit your responses. There is also more permanency the things you post today can be read for months even years. Another thing is the ability to search the forum for information. A forum generally is a slower moving more rich form of communication. It allows for and almost requires more patience.
They are both written forms of online communication consisting of messages and replies.
Imagine trying to type a response like that in a chat room.