The first sentence would likely come after:
I built a quantum computer in my spare time. After all, what is so hard about Quantum Field Theory?
Meaning the speaker fully understands the subject but not why others do not
To me the place of the "after all" is interchangeable - I do prefer the sentence above to this one:
I built a quantum computer in my spare time. What is so hard about Quantum Field Theory after all?
The second comes after a sentence like
All the nerds in my uni sweat day and night over their Quantum thesis'. What is so hard about Quantum Field Theory anyway?
Meaning the speaker does NOT understand Quantum field theory but doubts it can be hard or feels that his subject is much harder (Think Sidney Sheldon)
https://en.oxforddictionaries.com/definition/anyway (examples 1.1)
If you use anyway before the sentence, you are changing subject:
So I finally finished and went to watch TV. Anyway, what is so hard about Quantum Field Theory?