on that day和that day的区别主要在于用法及在句中代表的成分。on that day的中文含义为“在那天”，表示句子中的时间状语；that day的中文含义为“那天”，在句中可作主语、宾语、表语，也可作时间状语。

on that day意为在那天，在句中作时间状语。

例句：Are you free on that day?

那天你有空吗?

On that day, I decided not to let anger control me any longer.

那天,我决定不再让愤怒控制我。

And we were out of town on that day.

那天我们一起出城了。

And data collected on that day is also of great use in the long run.

而且双十一收集的数据长远来看也大有用处。

We don't exactly know what happened on that day

我们不知道那一天实际发生了什么

On that day last August, they were rather the opposite.

而在8月底的那一天，我看到的根本是敌对的反面。

Are you going to attend the seminar on that day?

你打算参加那天的研讨会吗？

On that day, they let us out of school basically as soon as we got there.

那天我差不多刚到学校学校就让我们回来了。

that day意为那天，在句中可作主语、宾语、表语，也可作时间状语。

例句：The memory of that day was indelibly printed on his brain.

那天的记忆永不磨灭地深印在他的脑海里。

She gave us her version of what had happened that day.

她从她的角度向我们描述了那天发生的事情。

He's borne me a grudge ever since that day.

从那一天起他便对我怀恨在心。

For once, fortune was on our side: the sun shone that day.

只有那次我们运气好，那一天阳光普照。

He was exulting in a win at the show earlier that day

他正在为当天早些时候在节目中胜出而欢欣鼓舞。

That day marked the beginning of a 20-year estrangement.

从那一天开始了长达20年的互不往来。