Not the most precise answer but, to me, when something bounces, it's already in motion, comes into contact with a surface and, as a result, moves in the opposite direction. When something springs, it is not already in motion and there is more of a feeling of a sudden release of energy.
So for example, if I bounce a ball off the wall, the ball first moves towards the wall, then hits the wall, then moves in the opposite direction. More abstractly, as well, if I bounce an idea off of someone, I first convey my thoughts to that person, the person considers the idea, and then gives me feedback.
If something springs into the air, it starts at rest, maybe compresses first, and then the movement into the air is sudden. Similarly, if you spring into action, you suddenly start doing something (in order to deal with a situation). Or if you spring to the rescue, you act quickly to help someone.