Welcome.

You probably did put them together too fast. But even with proper seperation acceptance takes time. Cats are usually NOT best friends instantly. Your older kitty needs times, possibly MONTHS.

The fact that your new kitty is young and playful does not help the situation. You may find that he pesters old kitty to distraction. Make sure he has a lot to keep him occupied so he doesn't get on old kitties nerves (so to speak).

Just don't force them together thinking it will help. If old cat is not fighting with new cat then they can adjust on their own time.

Good luck and don't worry so much. I have introduced litterally dozens of cats to one another and they always find a way to live with their house mates.