"Throwing at" means throwing a missile (usually forcefully) in the direction of somebody with the intention of the missile hitting the person. It's either antagonistic, or playful (with someone you know well). It's violent to throw a weapon, rock, chair "at" someone. Doing so is likely to cause injury, of course. But if a good friend is teasing me about something then I might crumple up a piece of paper and throw it at them (for fun and to tell them to shut up).
"Throwing to" means throwing a missile (sometimes forcefully, depending on the distance it has to travel) in the direction of somebody with the intention of the person catching the missile. You throw a ball to a team-mate during a game, or you throw a bag of chips across the room to your friend if you can't be bothered walking over there. :)