Here's a few examples to illustrate the difference in meaning and usage of 了解 and 理解.

1) 我不了解他。I don't know him very well.

2) 我不理解他。I don't understand him.

The first example relates to how familiar or well acquainted you are with someone. e.g.

A) 那你觉得他喜欢我吗？So, do you think he likes me?

B) 我不知道，我不是很了解他。I'm not sure, I don't know him very well.

The second one relates to the motives as to why someone would be the way they are, or why they did whatever they did. e.g.

他说他喜欢我，但是我不能理解他为什么还要和别的女人约会。He said he likes me, but I don't understand why he still dates other women.

Additionally, the word 理解 can be used in reference to a topic such as books, movies, situations, etc, to denote understanding (or lack of understanding) of the said topic. If you read a novel but didn't really understand its message or themes, you could use 理解 to express that.

1) 我不是很理解鲁迅的小说。I don't really understand Lu Xun's novels.

2) 我不是很了解鲁迅的小说。I don't really know much about Lu Xun's novels.

3) 我不太了解他的情况。I don't know much about his situation.

4) 我不太了解政治。I don't know much about politics.

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Another example to differentiate 了解 and 理解 would be this:

1) 我不是很理解马克思的经济理论。I don't really understand Marx's economic theory.

2) 我不了解马克思的经济理论。I don't know much about Marx's economic theory.

In 1), the person has probably studied, or at least been exposed to Marx, but has failed to get a good grasp of his theories.

In 2), the person probably has seldom - if at all - been exposed to Marx, and therefore doesn't know much about his theories.