"Please help me."
Both mean "help" since they contain 帮 (bāng, bang1, "to help"), and many use them interchangeably. There are slight differences in connotation, though not everyone necessarily agrees on what they are. In my experience...
帮忙 is less formal and more about distributing labor.
You'd say this if you need "an extra hand" to make everyone a little less 忙 (máng, mang2, "busy"). Since it's less formal, you'd use it to describe lifting crates together but not outsourcing professional work.
帮助 is more formal and more about support.
The 助 (zhù, zhu4, "assist/support") is the same as in 助理 (assistant/secretary). When someone revives you in a multi-player game, that is better described as 帮助. This is also used for more formal documents requesting aid.
So, in summary:
• spreading out one task ~ 帮忙
• assisting/supporting ~ 帮助