AVI and MP4 are just containers. They can hold a number of media streams-- usually 1 video stream and 1 stereo audio stream. Those streams can be encoded in different ways, called "codecs". Codecs include MP2 (on DVDs, which are also containers for media streams), H.264, DivX, and some others. So when the TV says it can play AVI files, it can interpret AVI files and play back some potential stream types contained in those files, but apparently not ALL media stream types.
There IS a type of copy protection to stop ripped DVDs from playing.
You might see if you can use Handbrake ( http://handbrake.fr/
) to convert to something that WILL play.
Downloaded TV shows are usually encoded with DivX or Mpeg 4.
But yeah, a cable to go from your laptop to your TV might be less trouble.