The verb to solve is generally used to mean find a solution – for example an answer or explanation. We solve a problem, something with a logical or complete answer.
The verb to resolve has a number of meanings, one of which is to deal with conclusively – that is, to settle something, effectively to finish it in an acceptable way.
The conclusion reached with resolving something may be one of many choices, and it may not please everyone, but it concludes the problem, finishing it.
The conclusion reached with solving a problem, however, suggests the correct and definite answer has been found.
To generalise, we tend to solve something objective, and resolve something subjective. That is, problems that require solving have a factual, simple (typically right or wrong) solution; problems that we resolve have a debatable, flexible solution.
The word resolve also means to make a formal decision about something, usually by a vote.
