BuddyPress
BuddyPress is really a plugin to wordpress adding basic social network features to the blog cms foundation. I've worked a lot with wordpress recently so that probably swayed my decision to try BuddyPress.
Features:
Out of the box buddy press offers
- Activity streams
- Extended Profiles
- Friend connections
- private messaging
- groups
- forums
- albums
Elgg is an open source social network project and is specifically built to be a social network.
Features
- Profiles
- Activity Feed
- Groups
- Pages for users and groups
- Blogs for users and groups
- Group discussions, bookmarks and calendar
- Messaging system
The main difference between the two is how you want to generate content. A BuddyPress system is great for having the administrator publishing a blog and maintaining website pages with user content primarily in the forums. On the other hand Elgg doesn't really have any tools for an overall blog and website but has lots of ways for content to be published by users and user groups in addition to group discussions (forums).
BuddyPress is easier to skin especially if you are familiar with wordpress themes, and may have a slightly larger support base because of it's close ties to wordpress. I'd say use BuddyPress for a site that publishes content and wants a community discussion. Use Elgg if you want to empower your community more.