For me, Leave me do it is unacceptable. It must be leave me to do it.
The difference is that leave me to do it implies to me, 'Don't bother about it; I'll do it' . Let me do it is fundamentally asking for permission, though it may be used as an offer: let me carry your bag for you.
If let me do it is uttered in the right context with the appropriate intonation, then there isn't a great deal of difference between that and leave me to do it.