difference between "can do nothing" and "cannot do anything"

Solution 1:

There is no significant difference between the two, as they both carry the same meaning.

If you can do nothing, you cannot do anything.

If you cannot do anything, the only thing that you can do is nothing.

Solution 2:

In common conversational understanding of the two phrases, there is no difference.

However, a difference can be parsed under analysis.

I cannot do anything to help you.

This means that I am incapable of doing anything to help you.

I can do nothing to help you.

While this can (and often does) mean the former, it's also possible that it can be taken in a different way.

For instance:

"What you are able to do for me in this situation?"
"Why, I can call an ambulance—or I can do nothing. Which would you like?"

In other words, it can be taken to mean that somebody is capable of doing nothing as one of several different possibilities.

This particular meaning would normally be expressed by emphasizing can, so as to indicate that it is being used in a less common way that the normally assumed meaning of the phrase.

However, there is another use in which this meaning is explicitly given:

"I'd like you to just sit here and do nothing. Can you do that for me?"
"Yes, I can."

Here, the directive of doing nothing (as opposed to something) is given in the question itself.