Chasing

See also: chase (if the game is about a chase)

Chasing generally has the player trying to catch up to someone running away. Tailing on the other hand tends to involve slight stealth, with player being required to maintain some distance and letting the target reach their goal but also making sure you see where that is too. The only real difference (in gameplay terms) here is that one requires "stealth, the other does not. Both usually require that you don't get too far behind or lose sight of your target for too long lest you fail.

This is NOT for some random enemies running away from you.