They're not quite the same.
Let's say that 'then' refers to a concert that you went to last year.
"I've been a fan of music since then" means that your transformation from non-fan to fan occurred at that moment in the past. When you were at that event you suddenly became a music fan. Your newly-found enthusiasm for music began at that concert.
"I've become a fan of music since then" means that the transformation happened during the period between the concert and now. Perhaps the concert made you start thinking about music, and then afterwards you began listening to more music and gradually became a fan at a later point.