When they show the meaning "sorry", they don't have difference . but sometimes "不好意思" has another meaning ,that is shame. for example,he felt "不好意思" ,so many people staring at him。（这么多人看着他，他觉得很不好意思）

"不好意思(excuse me)" and "对不起(sorry)" both mean apologize, but “对不起(sorry)”is deeper level than "不好意思(excuse me)".

对不起is former than 不好意思,and it means appologize seriously ,but 不好意思means maybe it is my fault,and i say sorry to you .

楼上说的都很对，我补充一下： “我不好意思说啦”这里的“不好意思”就是害羞的意思，表明这件事难以启齿（I am embarrassed to say ）

Actually even for me, a chinese, I cant tell the difference. But I often use 不好意思。I think compared with 对不起，the tone is lighter。 And 对不起 is more former than 不好意思.