There is a lot of overlap in the meaning. "못하다" can mean "you can't do something because you haven't learnt how to do" while "할수없어요" can have the same meaning as "못하다" or it has sense of "you are not able to do something because you're not in the mood for some other reasons" for example, 저는 수영를 못해요 = I can't swim(because you haven't learnt how to swim) 저는 수영을 할수 없어요 could mean... 1) I can't swim(because I haven't learnt) or 2) I am not in the mood to swim for some reasons like..... you didn't bring your swimming suit or you're tired, etc. (in here, you know how to swim^^/) \^o^/

It all depends on the context. 못해요- it has a meaning of " can't do" where as 할수없어요- has a possibility of "can do" but ,the condition doesn't seem to allow to do so. If you say 못해요 in certain circumstance, it can be very abrupt and not allowing any further chance of asking. for example: 이것좀 해 주시겠어요? 못해요. in this case, 못해요 has a meaning of " i won't do it" which has stronger meaning than can't.

Yeah, you're right. Both sentences has almost same meaning. little different... and Koreans usually say 'Mot hae yo.'

The meaning is the same. If there's a connotation difference, I haven't been able to tell