I've been playing with Buzz since it launched and I have a few interesting observations from my experiments. The first is its (and Facebook's) largest failure: Buzz is not for celebrities. Twitter is still the single best venue for celebrities. Why? Because replies gain little visibility from their association with a celebrity post. On Buzz and Facebook a reply associated with a celebrity posting is immediately visible to everyone who follows that celebrity and that creates enormous incentive to post mindless drivel and spam in response to any celebrity posting. This might be improved if Buzz had two kinds of "following": one for friends and one for celebrities and other people you don't know directly. The primary difference I'd be looking for would be hiding replies and not bumping an item in my list when it's replied to if it's not a friend.
Next up: local Buzz search is really nice. Be it on Google's map software or stand-alone, the ability to see what people nearby have been saying is huge. Of course, it will require more adoption to see its potential realized, but it's a great start. Some use cases include searching for a nearby restaurant that people like, finding the best spots at a public event and asking "the neighborhood" simple questions like where people like to do their laundry. If I move to another town someday, I'll certainly be doing lots of buzzing right off the bat.
Integration with other services: mixed bag. Until Google solves the problem of messages from other services (Reader, Yelp, etc) being duplicated when people reply to them on the remote service, Buzz will be a poor place to comment on shared comments and articles from other services. Still, the integration with these services is promising. I was able to do much of what I'm now doing with Buzz in Reader by following RSS feeds from other services (e.g. following my friends' YouTube feeds). But Buzz makes it easier and much more social.
I think Buzz is probably here to stay, and while I'm sure Facebook will continue to do well, it's clearly the primary target of Buzz and it will almost certainly feel the heat.