男人 and 女人 draw attention to gender

This difference is far less concrete than the one above, and is probably quite debatable. However, it's often that case that the words 男人 and 女人 (nǚrén) in Chinese are used when the person's gender is particularly relevant. Otherwise you just say 人.

In other words, in English we often refer to people as “a man” or “a woman”, and less often as “a person”. In Chinese, though, you can often just say “a person”.

For example, in English you would probably say “he's a very popular guy”, and less commonly say “he's a very popular person”. Again, this may differ from person to person. In any case, it's usually better to say “person” in Chinese unless the gender is very relevant.

Some examples:

他是一个备受欢迎的人。

Tā shì yīgè bèi shòu huānyíng de rén.

He's a very popular guy. (person)

她是个很有成就的人。

Tā shìgè hěn yǒu chéngjiù de rén.

She is a very successful woman. (person)

前门有个人来找你。

Qiánmén yǒu gèrén lái zhǎo nǐ.

There is a man / a woman / someone to see you at the door.

Again, you could of course say “person” in each of the above sentences in English. Happily, this practice is becoming more and more common in English as people put less emphasis on gender. The point here is that it's usually better to use 人 in Chinese.

The situations where you should use 男人 and 女人 in Chinese are when you really are singling out men or women and not including both genders. Some examples:

女人一般比男人长寿。

Nǚrén yībān bǐ nánrén chángshòu.

Women usually live longer than men.

男人工作时受伤机率比女人的大。

Nánrén gōngzuò shí shòushāng jīlǜ bǐ nǚrén de dà.

Men are injured at work more often than women.

女性的平均工资仍然没有男性的高。

Nǚxìng de píngjūn gōngzī réngrán méiyǒu nánxìng de gāo.

The average salary of women is still not as high as that of men.