sorry是一个形容词。
例句
I am sorry.
He is sorry for doing....
Sorry about ...
I feel sorry. (modal verb + adj)

sorry有时候也有遗憾的意思

I am sorry to hear that. 很遗憾听你这么说。-比如你听到别人的糟糕的经历，你叹息一下表示同情。
I feel sorry for him. 我很同情他。



apologize 是动词

I apologize for ....
No need to apologize.
还有一首歌 It's too late to apologize.

apology是他的名词形式
My sincere apology for ....

什么时候用sorry/apologize?
我的感觉，你中文里面觉得要说 “对不起”的时候 用sorry
你觉得你要说，我道歉，或者 我要表示我的歉意，你就说 I apologize for this and that... 或者 I would like to express my apology ....