All three mean “to help” so, what’s the difference?
帮忙 [bāngmáng]
To help.
More informal.
Typical structure: 请 [qǐng] /让 [ràng] + 帮忙 + verb
爸爸让你帮忙做饭 [bàba xiǎng ràng nǐ bāngmáng zuòfàn] Dad asked you to help him make lunch
帮 + somebody + 一个（大）忙: to lend somebody a hand
明天我要帮妈妈一个忙 [míngtiān wǒ yào bāng māma yī gè máng] Tomorrow I want to lend my mother a hand
Since it’s a separable verb, you can’t add an object or person afterwards.
❌ 帮忙做饭 [bāngmáng zuòfàn]
❌ 帮忙妈妈 [bāngmáng māma]
It can be used by itself.
你快来帮忙！[nǐ kuài lái bāngmáng] Come and help, quick!
帮助 [bāngzhù]
To help somebody do something.
More formal.
It can be a verb or a noun.
谢谢你的帮助 [xièxiè nǐ de bāngzhù] Thank you for your help
As a verb, the typical structure is: 帮助 + somebody.
Although not technically incorrect it doesn’t usually go alongside other verbs, just with the person receiving the help.
你可以帮助我吗？[nǐ kěyǐ bāngzhù wǒ ma] Can you help me?
帮 [bāng]
To help somebody do something; to do something for somebody.
Most common in day-to-day speech.
Typical structure: 帮 + somebody + verb + noun
帮我做作业 [bāng wǒ zuò zuòyè] Help me to do the homework
你能帮我开门吗？ [nǐ néng bāng wǒ kāimén ma] Could you close the door for me?
It is usually repeated or “reduplicated” 帮帮.