"帮" (vt)"请帮我开门!qing bang wo kai men!" Help me to open the door, please!
"帮助"(v) "帮助别人是我的乐意bangzhu bie ren shi wo de leyi" Helping others is my pleasure
(n)"谢谢你的帮助xiexie ni de bangzhu" Thank you for your help (formal)
"帮忙"(n)"谢谢你的帮忙xiexie ni de bangmang" Thank you for your help (informal)
(verb phrase)"只有他可以帮你的忙zhi you ta keyi bang ni de mang" Only he can help you
The difference between "帮" and "帮助" :
"帮" one syllable
"帮助" two syllable
sometime Chinese care about the symmetric in a phrase or in a sentence
so you can use "帮" for the object with one syllable (帮我好不好?) Can you help me?
and "帮助" for the object with two syllable (互相帮助) Helping mutualy
The differences between "帮助" and "帮忙":
When it is a noun, "帮助" use for normal situation, "帮忙" use for innormal situation
and another difference is "帮助" is a verb: 帮助 + obj, and "帮忙" is a verb phrase: 帮 + obj + 忙