I always use like this: 对不起 is used when the situation is relatively formal, or the person you are familiar with and you want to show your sincery. 不好意思 is used when it's just a sightly mistake (like I almost stomp your foot but not actually have done it, if you did，对不起will be better)

Hi Qays 不好意思bu hao yi si 是抱歉，但并没做错。/ 程度较轻的道歉 A lesser apology 对不起dui bu qi 则是做错事。/ 正式礼貌的道歉 A formal and polite apology 不好意思bu hao yi si同时can also mesns还能表示execuse me的意思

I remember when I was little , I felt I did something wrong and I would say “ sorry” , but one time my sister told me that don’t say dui bu qi ,say bu hao yi si . I was quit confused for a while , but now I think the different is . When you did something wrong and that effect to others serious you say dui bu qi . If some small mistakes or even sometimes u kinda wanna people get notice of u , you would say bu hao yi si

Basically duibuqi means I am sorry, while buhaoyisi is sort of Excuse me. If you faithfully make an apology for your mistake, duibuqi is more acceptable.

sometimes I will use both at one time to show that I really feel sorry and I'm trying to be as polite as I can. For example, 对不起，xxxxx，实在不好意思。 or 不好意思，xxxxx，实在抱歉/对不起。