In most cases, they are the same. In my humble opinion，they are slightly different at these cases： 1. 理解 lijie is a bit more literary, while 明白 mingbai is a bit more colloquial. 2. 理解 has a sense of empathy, for example, 我不赞同他的行为，但我能理解他。 3. 理解 is used when comes to understand something a bit difficult, like science or literature, means comprehend / comprehension, fox example, 理解能力 the ability of comprehension.

From a Chinese friend when I asked this question a long time ago: 理解 - Understand (formal). You are learning something that you don't understand. After thinking hard about it, you 理解了 it. Eg. a teacher teaches his students a theory. After that, the teacher asks “你們理解了嗎？” 瞭解：To know Some information that you didn't know before. Eg. 通過對這篇文章的閱讀，他瞭解到中國是一個很大的國家。You can't use 理解 here. 明白 - Understand (informal) Teachers also ask students “你們明白了嗎？” Eg. 你在给Peter写信吗？不行，你现在还不能给他写信，因为（some reason），你明白了吧？ 懂 - The same as 理解 and 明白 and can be used in both informal and formal situations 領悟 - After a very long time and some hard thinking, people 领悟到了 stuff. Eg. his father told him something when he was a child. At that time he couldn't understand what his father meant. Now he is 40 years old and he finally gets it. ----- 他領悟到了他父親当年說的那句話的意思了。

Thank you very much!

Actually they have no differences in spoken Chinese.You can both say "li jie" and "ming bai" when you understand, and you can both say"wo bu li jie" and "wo bu ming bai" when you don't understand. But "li jie " seems more formal, especailly the word is not used alone. There is a special situation that people can only use "lijie" not "mingbai" is that " xiang hu li jie".We cannot say "xiang hu ming bai".The meaning of "xiang hu li jie" is "know each other".You can view it as a fixed usage. The same as "xiang hu bu li jie" .Another difference is that "li jie" can ＋somebody，but "ming bai" can't. For example,you can say" wo li jie ni",but you should not say "wo ming bai ni".Both "li jie " and "ming bai" can ＋something. For example, "wo li jie ni de yi si"＝"wo ming bai ni de yi si",and the former one seems more formal.