I know that he is rich 和I know he is rich 有什么不同？

其实两个都是从句，I know (that) he is rich. that 可以省略的。

宾语从句中that一般情况可省 ,但that在引导宾语从句时也并不是任何情况下都可以省略。在以下情况下，that不能省略。

1． Everybody could see what happened and that Tom was frightened.(and连接两个宾语从句，that宾语从句放在and的后面时，that不能省略。)

大家都会看出所发生的事情并知道Tom非常害怕。

2．I know nothing about him except that he is from the south.(that引导的宾语从句作介词宾语时，that不能省略。)

对他我一无所知，只知道他是南方人。

3．That he ever said such a thing I simply don’t believe. (that从句位于句首时，that不可省略。)

我简直不相信他曾说过这样的话。

4． We decided, in view of his special circumstances, that we would admit him for a probationary period. (主句谓语动词与that从句之间有插入语，that不可省略。)

I know that he is rich \
从句.
I know he is rich
肯定句.

没区别，都是正确的。
that 可省略