The two products are alike in name and the fact that they're social. But that's about it.
Google Buzz lets you be social from Gmail. You can update your status there and share photos. (Yahoo! Buzz is more akin to social bookmarking and trending topics.)
Who do you share Google Buzz with? Gmail contacts, for one, but you can also send your updates to Twitter as well. Your Gmail contacts can respond to your updates, using the @ symbol much like Twitter and now Facebook use. Your Gmail inbox will be used to push notifications to Buzz users.
Buzz is rolling out, so it may take a few days before you get access. In the meantime, check out this vid to see how it works:

