"problem solved" is the phrase we use and it sounds more naturla than "problem settled". "problem settled" does not make proper sense. "the matter has been settled" implies that a situation has calmed down or been resolved or solved. but you can say the "the problem has been settled" it makes sense but we just use "solved" when we talk about a problem.

To settle : a matter or substance comes to rest Solved : A problem has become a non issue or is resolved There are times when these overlap as you see. It could be that a matter between people could either be solved or settled. In these cases it is better to have a disagreement solved AND settled. This would mean that there is no longer a problem between two groups AND the groups are not going to argue about it. You will see soon how these are similar, yet not the same... When a problem is settled, there may still be sore feelings. If something unfortunate (for instance a poorly chosen comment) where to be done or said, a settled issue can be re-stirred and once again people may be volutile. Further more, even if there is a solid agreement to get along, there may or may not still be an unresolved problem. What if the argument was about to fix brakes at the car care place or a dealership. The parties agree to not bring it up and simply drop the issue. It is settled... Till someone crashes? On the other hand, if the problem is solved (the brakes are fixed) does that always mean we will not argue over it? Not in my experience! So we can see that settled is not always solved and solved is not always settled.

The problem has been solved = we found the solution we were looking for or needed. The matter has been settled = Usually means we have reached a resolution with another person or party.