The simplest difference is this:
Question 1 is not as easy as question 2.
This means question 2 is easy, question 1 is harder.
Question 1 is not easier than question 2.
Question 1 may be as hard as question 2, or it may be harder.
So in the case that question 1 is certainly harder than question two, both phrases are correct. But in the case that both questions could also have the same difficulty, only sentence 2 applies.
More importantly, there is also a different implication about the difficulty of the questions:
Question 1 is not as easy as question 2.
Someone may think that question 2 is too easy, and suspect the other questions are easy too. You then tell them that question 1 is in fact not as easy as 2.
Question 1 is not easier than question 2.
Here, someone probably thinks that question 1 is too easy, but your reassure them that the question is certainly not easier than question 2.