1、意思上的区别

keep up with：紧跟；跟上，不落人之后；齐肩并进。

catch up with：赶上；追上；和…算旧账；使…最终尝到苦果。

2、用法上的区别

keep up with ：是有关于你学业／工作的水平而言。（褒义）

catch up with：是有类似于追上的意思。（中性）（或者有自贬意思）

例句：

①、keep up with

She was skipping to keep up with him.

她连蹦带跳地走着，好跟上他的步伐。

She began to walk faster, but he lengthened his stride to keep up with her

她开始加快步子，但他加大了步幅跟着她。

②、catch up with

I couldn't catch up with him.

我撵不上他。

There's still time to catch up with them.

还有时间追上他们。

We'll do our best to catch up with the advanced world levels.

我们将努力赶上世界先进水平。