Why can't I delete movies from my iPhone 4, either from the phone or through iTunes?

While not a SuperUser-compatible answer, since the computer interface is missing, I just did the following:

  1. Open the iPod app
  2. Go to Videos
  3. Swipe over a video from right to left
  4. Press Delete
  5. Confirm
  6. Done!

No computer needed. Also works for music