couple和pair都可以表示“一双、一对”，但它们的用法存在着区别如下：

(1)couple：指任何两件同类的东西/事物，还可以表示夫妻二人。

a couple of cats  两只猫

a couple of shoes两双鞋子

The couple have no children.  

这对夫妻没有孩子。

They were an odd couple.  

他们是奇怪的一对。

She had hoped the couple would put on a show of unity.  

她曾经希望那对夫妇会表现出和睦相处的样子。

The couple had helped in the hunt for the toddlers.  

这对夫妇曾帮着搜寻那些幼童。

He presented us with a couple of jade lions.
他送给我们一对玉狮子。

Her cheeks began to fill out after she'd been in the convalescent hospital for a couple of months.
在疗养院养了两个月以后,她的脸渐渐变圆了。

We concede a couple of points in your contentions.
我们承认你的论点中有两点是正确的。

(2)pair：指两件不能分开使用的东西；它们可以相连，也可以构成一件东西的两个部分。

a pair of shoes 一双鞋子

These two cups will make a pair.
这两个杯子可以配成一对。

This pair of shoes fits me well.
这双鞋对我很合适。

Another pair of shoes is needed.
还需要一双鞋。

There are two pairs of shoes under the bed.
床底下有两双鞋。

I found a couple of socks in the bedroom but they don't make a pair.

我在卧室找到两只袜子，但配不成对。

You are lucky couple.

你们真是幸福的一对。

The old man wears a pair of glasses.

老人戴着一副眼镜。