Which sentence is correct?
How different they are
How they are different
I think both are right because one is intended as the meaning of "degree" and the other as the meaning of "way".
What do you think about this?
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(this is)... how they are different
how they are different ... (is bla bla bla)
This sounds like it should be part of a sentence explicitly explaining the differences between the two. (although it sounds a bit clumsy admitedly)
This explains how different they are.
How different they are!
This could be used much the same, however, it could also just be a statement remarking that there is a large difference (especially adding the exclamation mark).
How has a different meaning when it directly modifies an adjective.
How different (in any context, I think) means "how big is the difference".
In other contexts, how means "in what way".
So, as direct questions:
How are they different? means "In what way or ways are they different?", but
How different are they? means "How great is the difference between them?"
Exactly the same applies to indirect questions such as "She asked me how they were different" vs "She asked me how different they were".