I have both. The Mac Mini acts as a audio and video server, but controlling it through keyboard and mouse is a pain from the couch. That's where something like the Apple TV works well, because it's purpose built to make browsing for shows, movies, YouTube, Internet Radio, etc. very straight forward and easy in the living room.
I recently purchased a Western Digital TV Live, a device which competes with the Apple TV. Same price, similar access to services, but the WD Live also lets you use your computer as a fully enabled source for content. If, for example, you have ripped a DVD and stored it on a drive through the Mac Mini, you can now play that ripped DVD from the WD Live device. Pretty much any file format, any number of sources.
By contrast, Apple TV will only have access to Apple approved content sources on the Internet, and iTunes imported content on your Mac Mini.