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When I am thinking how to best explain my idea, I got the following doubt

I do not understand why...

I can not understand why...

I feel both sentence pattern give a same meaning. But again I feel, if they are equal why there are two ways, so that I feel these two should give two different meanings. But for me, I can not figure out what is the main difference of using ... do not ... and ... can not ...

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In the context of understand, cannot and do not amount to much the same thing: I cannot points a little to the effort you have made to understand, but usually if you don’t understand something it is because you can’t understand it.

In other contexts, however, cannot and do not may mean quite different things.

I cannot read Cicero probably means that you don’t know enough Latin to read Cicero.
I do not read Cicero means you choose not to read Cicero—you probably find his works boring.

You should not be disturbed by the fact that two expressions mean approximately the same thing. In fact, there are many more than two:

I am unable to understand why.
I fail to understand why.
I am at a loss to understand why.
I cannot conceive why.
I have no idea why. ... and so forth.

There are many ways of saying practically everything. I am confident you can point to instances in your own language.

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