Learn English – “the little money he had” versus “what little money he had”

Before considering them just as noun phrases I want to look at them as complete sentences.

Doing so, only one of them is idiomatic.

For the sake of argument, I will turn them into emphatic sentences:

✘ The little money he had!
✔ What little money he had!

While the first is not technically wrong, it's not something that would normally be said outside of a larger context.

I think a comment you made is correct. The on its own normally acts as an article, something that introduces a noun as the subject. Even with the exclamation point, the first sentence sounds odd. On reading it, the inclination is to ask, "What about the little money he had?"

But in the second sentence, the use of what indicates that the sentence is about making an emphatic point that forms a complete thought that doesn't require further exposition.

Further:

A green tree.

This reads like a noun phrase rather than a complete sentence with a finished thought.

What a green tree.

This reads like a complete sentence, declaring how green the tree is.

Both of these could also be considered elliptically:

What a green tree (it is).

This makes sense.

A green tree (it is).

This also makes sense, but sounds unnatural. It's an elliptical version of Yoda-like speech that people aren't used to hearing. Written in full, we can make sense of it. But we don't normally assume something is elliptical when the full version is nonstandard.

Interestingly, the situation changes if a slight alteration is made:

✔ Oh, the little money he had!
✔ Oh, what little money he had!

The first sentence, although I would say it is still less common than the second, becomes something no longer as rare (and, hence, odd) when it's preceded by an interjection.

The use of the interjection in the first sentence signals the reader that what follows is not just a noun phrase but a functionally complete sentence—one whose entire meaning is emphatic declaration.

As noun phrases as part of a larger sentence, as you'd originally written your question, I don't think there is any essential difference between the two.

He spent the little money he had.
He spent what little money he had.

To me, both of these sentences mean the same thing.

However, although I read both as indicating that he had little money, I could add an additional interpretation to the second sentence: shock over the fact that he had little money in the first place.