The word for "help":「助ける」と「手伝う」 tasukeru vs. tetsudau

Both words mean "help" in English. However,　手伝う "tetsudau" refers to helping someone with the same task. 助ける "tasukeru" can be used in place of 手伝う in some cases. However, unlike 手伝う, it can be used to refer to saving or rescuing someone (for this, 手伝う is NEVER used!) . From my encounters with the language, 助ける is more often used in this context of rescue than just helping someone complete a chore.

手伝う examples:

私は彼の宿題を手伝った。
Watashi wa kare no shukudai wo tetsudatta.
I helped him with his homework.

料理をするのを手伝ってよ。
Ryouri wo suru no wo tetsudatte yo.
Help me with cooking!

私は父が洗車するのを手伝った。
Watashi wa chichi ga sensha suru no wo tetsudatta.
I helped my father wash the car.

The basic pattern for 手伝う is AはBを手伝う, where A is the person helping, and B is the act that needs helping (not the person that needs helping). It is also possible to show the "direction" in which help is given by using ～てもらう、～てくれる、～てあげる.

私に手伝ってもらいたいですか。
Watashi ni tetsudatte moraitai desu ka?
Do you want me to help? (lit. Do you want to receive help from me?)

手伝う can also be made more polite by using the form お手伝いする.


放課後、お手伝いしてくれる？
Houkago, o tetsudai shite kureru?
Can you help us after school?


台所でママのお手伝いをしたの。
Daidokoro de mama no otetsudai wo shita no.
I helped mama in the kitchen.


助ける examples:

助けて！
Tasukete!
Help!

彼女は助けを必要とする。
Kanojo wa tasuke wo hitsuyou to suru.
She needs help. (Lit. As for her, help is considered necessary.)

誰が私を助けてくれるの。
Dare ga watashi wo tasukete kureru no?
Who will help me?

Note that in most cases, the pattern is AはBを助ける, where A is the person helping, and B is the person being helped.

助けに来る "tasuke ni kuru" is also commonly used. It means "come to (someone's) rescue".

誰も私を助けにこなかった。
Daremo watashi wo tasuke ni konakatta.
No one came to my rescue.

The direction of help can also be indicated by ～てもらう、～てくれる、～てあげる.

友達に助けてもらいなさい。
Tomodachi ni tasukete morainasai.
Get a friend to help you.

私は彼を助けてあげたい。
Watashi wa kare wo tasukete agetai.
I want to help him. (Lit. I want to give him help.)

Reference:
jisho.org
Miura, Akira. Essential Japanese Vocabulary, Tuttle. See Foreword and entry on 助ける.