They have very similar meanings when used together with the word "problem".
However, "solving" is often used when the problem is academic, mathematical, or mechanical; whereas, "resolving" tends to be used when the problem relates to human differences and/or emotions.
Thus, "I couldn't get the top off the can, but I solved the problem by drilling a hole in it", compared with "We always fought over who should get first pick of the chocolates, so we resolved the problem by buying a box each".