Question

the reason why 和 the reason that怎么用？有什么区别

Answer 1

the reason why中的why是关系副词，当先行词the reason在定语从句中作原因状语时，可以用the reason why。
I know the reason why she was late.
我知道她迟到的原因。
That is the reason why my teeth are bad.
这就是我牙齿不好的原因。
当先行词the reason在定语从句中作主语或宾语时，用the reason that，that在这里是关系代词。如：
That is the reason that he gave me for taking this action.
那是他向我提出的采取行动的理由。
That is the reason that was given for not going to college.
那是被提出的不上大学的理由。

Answer 2

定语从句句型the reason why 和 the reason that的区别主要如下：
the reason why，……的原因。后面的定语从句描述的通常是该原因引起的后果，why是关系副词，在从句中作原因状语。
the reason that，……的原因。后面的定语从句中若没有介词for，则从句通常是对这个原因本身而非其后果进行描述，that是关系代词，在从句中作名词性成分；后面的定语从句中若有介词for，则通常可以与the reason why的句型互相转换保持句意不变，that是关系代词，与for合在一起在从句中作原因状语。
例句：
⒈The reason why the Mayans disappeared remains uncertain.
玛雅人消失的原因至今不明。
⒉The reason you explained to me is unpersuasive.
你给我的理由没有说服力。
⒊This is the reason why I appointed him as my heir.
这就是我指定他为继承人的原因。
=This is the reason that I appointed him as my heir for.
=This is the reason for which I appointed him as my heir.
另外，the reason that 引导的不一定是定语从句，还可能是同位语从句等。
例句：
⒋He gave me the reason that his car broke down.
他给我的理由是他的车子抛锚了。