If you transmit, copy or otherwise distribute copyrighted materials without the explicit consent of the copyright holder, you are in violation of the US copyright law, and could be sued by the copyright holder and be subject to criminal charges as well. It doesn't matter where you "get" the videos from. If you include a copyrighted video on your tv show, you must purchase a license.
The "free publicity" angle is fine, if you receive permission from the copyright holder. However, the 'singer' could make the same case from their point of view (e.g. You're using their videos to get free content to promote your show). I can guarantee that if the 'artist' is signed to a label, that label would be all over your case for using the video without permission. If their marketing plan for the artist doesn't include your show, they won't be interested in your "free publicity". As for your "videos vs. movies" supposition, frankly that's ridiculous. Do you think a radio station could play one song from an album and say, "well, it's only one song, not the whole album".
Here's the deal. To legally use copyrighted video and audio:
(a) you need to purchase a mechanical license from the publishing company that owns the material
(b) if you use copyrighted audio as a background for your own video content, you'd have to purchase a mechanical license and a synchronization license for the video.
All this stuff applies to using the video and audio of major artists. If you are talking about unknown, unsigned bands, they're probably not making any money - and would love the publicity. Just make sure you have them sign a release for the content. If they're not making any money, they might look to 'sue' you - just to get some.