Well, there's a few things that come into it for me, the first of which is, "How sincere is the apology?" I remember, when I was in Year Seven, a boy in my class, who was a bully, drew a cruel caricature of me and the teacher asked to see it and he was angry and told him to throw it away and apologize to me. The boy walked past me and huffed an apology.
I was grateful to that teacher earlier in the year who, when I asked a question about something and the same boy said, "Duh!" asked another student for his ruler and said that he and the boy were going to have a little talk. He hit the boy across his upper arm and yelled at him.
In Australia, I hear some war veterans who keep saying over and over, "Oh, Japan hasn't apologized for the war." The truth is, however, Japan has apologized dozens of times over and how many more times do they have to apologize, but the media doesn't report upon it! For example, when then Japanese Prime Minister, Nobusuke Kishi visited Australia, in 1957, he gave a personal apology to then Australian Prime Minister, Bob Menzies.
I agree with Harold below that there's a distinction between accepting an apology and forgiving someone. Some people, including former POWs and war veterans, without knowing of any apology have said, "Let bygones be bygones."
But let's not forget, also, that even if you accept an apology, you still reserve the right to determine what, if any role that person has in your life. My father said of me 22 years ago last week, that if I get a set against somebody it lasts for life. Is that true? Well, yes and no. I mean, the person in question was the GP who I didn't like and what I wanted acknowledged there was, I didn't care who my parents see as a GP, and even if I forgave the GP I didn't like, I have a GP I like and trust now, and it doesn't mean that I would go and see the GP I didn't like. I mean, I have forgiven my ex-girlfriend for being a horrible person, but she and I aren't friends post-breakup.