TV resolution/compatibility
Whoever told you that was wrong.  "1080p" has 1920 horizontal pixels and 1080 vertical pixels.  "720p" has 1280 horizontal pixels and 720 vertical ones.  See this wikipedia page.  You would want to use 1080p whenever you want the highest quality your television is capable of reproducing.  The only reason you'd use 720p is if file size is a concern, or if your television doesn't really go up to 1080p.