If he is your best friend, I doubt he is going to break off his friendship with you over you failing one time to wish him a happy birthday or you not sending a gift to him.
I suggest that the next time you two meet up and go out to dinner or some event like a football game, as a way of showing that his friendship means a lot to you, you should pay for everything. This act of kindness should heal the wound of you forgetting to wish him a happy birthday.
If he still treats you coldly after that, then he is a pretty petty person and you trying to maintain a long-term friendship with him may end up causing you a lot of grief down the road.