To me, it sounds like he is saying 'sorry' to please you but does not agree he shouldn't bite.
My experience is with smaller children, and on the receiving end - DD has been repeatedly hit/bitten/etc by two kids she regularly sees.
What worked with them was being scared out of their wits by the mean mum (me) of the child they had the habit of hurting.
This was with knowledge of their mums, by the way. They said their sons didn't listen to them, and I was welcome to tell them off and see how that works.
They never came near DD again.
My point is that telling your son "it's not nice" is great, but of course he knows it isn't, because whenever he bites a child, he sees him cry and scream in pain.
What he doesn't have is a sense of consequences - what will happen if I bite again? Mum will give me a calm speech and I will say I'm sorry. Then I can go off and bite whomever I want.
What will work with your son, only you can decide. Perhaps have a chat with other mums and see what they suggest.