I've always been curious if there is a definite difference between the terms or if they just depend on the context of a problem.
I guess verify is more often used when the thing you need to prove is already given, but I don't think there is an essential one.
(I am German, maybe we use it a bit different)
You can verify a process by substituting given values.So when asked to verify there is no need to prove. But to prove something you need to arrive at the necessary conclusion step wise without any substitutions.
For example, a question might be asked as: Prove that $(a+b)^2 = a^2 + b^2 + 2ab$ and verify the statement with a=1,b=2.
So to prove the above statement,you follow the general step by step process by taking the LHS and arrive at the RHS. Where as to verify the correctness of the statement you just need to substitute the values and check for the correctness.
Hope the answer is clear !