There are two dimensions to their differences - the first the psychological dimension and the second the technology dimension. One of the major psychological differences is that you own your weblog - it is YOURS - and it represents a history of YOUR thinking - so you take pride in its ownership - something that does not make a lot of sense in a discussion forum.
On the technology front - Ray Ozzie sums up one of the major differences:
In traditional discussion, topics and their responses are contained and organized within a centralized database. The relationship between topics and responses is generally maintained in a manner specific to the nature of the database - that is, in newsgroups the messages might be related by Message-ID hyperlinks or crudely by title, in Notes they are related by the $REF hyperlink, and so on. Summary-level "views" are generated through database queries. And that has been the general architectural design pattern of public discussions for quite some time.From Architecture Matters: The Rebirth of Public Discussion by by Ray Ozzie
But blogs accomplish public discussion through a far different architectural design pattern. In the Well's terminology, taken to its extreme, you own your own words. If someone on a blog "posts a topic", others can respond, but generally do so in their own blogs, hyperlinked back to the topic's permalink. This goes on and on, back and forth. In essence, it's the same hyperlinking mechanism as the traditional discussion design pattern, except that the topics and responses are spread out all over the Web. And the reason that it "solves" the signal:noise problem is that nobody bothers to link to the "flamers" or "spammers", and thus they remain out of the loop, or form their own loops away from the mainstream discussion. A pure architectural solution to a nagging social issue that crops up online.