They're different programs.
Lightroom can function as a standalone program. It has basic Camera Raw adjustment features in it and maintains your images for you. You can also print from it. Its interaction with Photoshop is very limited, though. Mainly just an export option, and cannot add to PSD files without introducing layer flattening at some point in the process.
Bridge is a Photoshop companion program. It works alongside Photoshop as an image viewer (think Finder on steroids). It doesn't manage your images for you, but navigates through your custom organization. It has many quick shortcuts to useful scripts and automation tasks in Photoshop. By itself, Bridge isn't spectacular, but the way in which it incorporates back and forth with Photoshop is a great thing.