Even when a tape is rewound a small portion of tape is exposed at the bottom of the cassette. (Yes, on a video tape the tape is covered by that little flap, but if you look at the tape from the bottom you can still see the tape.) Since it is exposed, this portion of the tape can collect dirt, scratches and other damage. Leader tape prevents the damage from occuring on the potion of the tape that actually contains a recording.
Also, on VHS tapes in particular, a large part of the tape has to be pulled into the mechanism before it can start playing or recording. The leader allows for this in that you won’t have something recorded on a portion of the tape that will never be able to get to the playback head.
The same holds for reel-to-reel tapes too. The leader (or trailer, since reel-to-reels are normally stored “tails out”) is the outside of the tape when it is stored and you have to have enough tape to thread through the mechanism without losing recorded information. Reel-to-reels also had a habit of breaking at the ends and the leader allowed for the occasional loss of tape from the end without affecting the recording.