Actually there is no difference, it's just a matter of style. Indeed you can say both "Ha incominciato a suonare la musica!" and "Ha cominciato a suonare la musica!" and you are speaking a good Italian. A similar sentence could be "Ha iniziato a suonare la musica!". It has the same meaning but it is less used. As I said before, it's a matter of style.

Non c'è differenza, hanno lo stesso significato.