帮忙 means do someone a great favor.
帮 means do something for someone.
帮太太的忙 means do his wife a favor.
Do do and favor work together here or are they 2 separate words? The answer is both! why?
Do my homework for me please! (a little rude; don't you think?)
It's a great favor to me if you finish my homework! (a little arrogant and self-centered.)
can you do me a favor for finishing my homework! (sounds a little better, right?)
Thus, 帮忙 essentially means helping someone do something very important because 忙 means busy; ie, keeping someone occupied by doing this favor for you, and since Time equals money, it's a great deal.
So, use 帮忙 if you are asking a great favor, or if you are a polite person asking for a small favor.
And, use 帮 if you are asking a small favor, or if you are an arrogant person asking a big favor.
Here are some example:
- 帮我拿杯水过来。 Pass that cup of water to me. (Small favor, use 帮 is enough)
- 帮忙拿杯水过来。 Do me a favor, pass that cup of water to me. (Passing a cup of water will not make someone busy, thus it's too polite/inappropriate to use 帮忙 here.)