Stand still means to not move or remain stationary. Stay tight means to stay very close to someone. You might use it in a situation where you don't want to be separated from someone. Or, you can use it in sport where you stay tight to your opponent.

To stay tight or hang tight means "wait here" or "do not leave the area." To stand still means "do not move from the spot you are at." If you are at a friends house and they ask you to stay tight when they leave to get beer, it means don't leave the house. You can go into other rooms. If they say stand still, you cannot walk around or even sit. Hope that helped!