1. 帮助 usually used as a noun. E.g.:
多谢你的帮助。Thanks for your help.
没有他的帮助我肯定做不成这事。I can't make it without his help.
2. 帮助 can also used as a verb, but we always put the object that is helped after it. E.g.:
帮助他人让我快乐。Helping others makes me happy.
请你今天帮助我完成这项任务。Please help me finishing this job today.
As you can see, 帮助 is always followed by the object, "帮助+某人" or "帮助+某人+做某事", that's the basic pattern.
3. 帮忙 usually used as a verb, and the pattern is "帮忙+做某事". You can't put the object after 帮忙.
请帮忙抬一下这个箱子，它太重了。Please help me carry this trunk, it's too heavy.
他让你帮忙买早饭。He asks you to buy breakfast for him.
4. 帮忙 can also used as a noun, just like 帮助, but not that popular. We usually use 帮助 in this case.
I think they are same.
It means that you can use both words in most situation.
help me = 幫忙我 (it works) = 幫助我 (it works)
you don't help me = 你不幫忙我 (it works) = 你不幫助我 (kind of weird but it works to me)
I think either 幫忙 or 幫助 is [幫(verb) + something], so it is not necessary to distinguish them.
I was going to answer this question, but then I found a video that explains this far better than I ever could:
Please let me know if you're still confused after this video. I'll see if I can explain it another way.