Applications show as .app instead of in Alfred & Spotlight

Your Spotlight index might be corrupted – Alfred uses the same index as Spotlight.

If Photo Booth is the only item missing, you can try dragging it out of /Applications (to your desktop, say), then back in. You’ll have to authenticate as an administrator both times; Spotlight should pick up on it after that, and so should Alfred (after a small delay). If that doesn’t help, or other items are not found either, you will have to re-build the Spotlight index, either by following Apple’s official procedure, or via the Terminal command sudo mdutil -E /.

I believe that unless you have "Show all filename extensions" checked in Finder's Preferences, the .app extension is hidden by default. If you were to look at Sparrow in Finder, do you see the .app? If so, open the Get Info window for it (Cmd+i while highlighted) and make sure "Hide extension" is checked. You would have to check each app that is showing its extension manually.

Another thing is that if you have more than one of the app, I think it will always show the extension in Spotlight.

If none of that works, you can always try rebuilding your Spotlight index: $ sudo mdutil -E /