May through September is the most comfortable time to visit. The weather is mild rather than spectacular — this is northern Germany — but the days are long and the old town is at its best when people are actually sitting outside it. The Schlossgarten and the parks around the city walls are pleasant for walking in summer, and the outdoor market on the Marktplatz is at its fullest.
The Friedensfest in late July commemorates the Peace of Westphalia, signed in Osnabrück and Münster in 1648, and the city takes it seriously with events around the Rathaus and the old town. December brings a Christmas market to the Marktplatz that is worth a detour if you are passing through the region; the Gothic backdrop of the Marienkirche and the Rathaus suits it well. February is the low point: grey, cold, and quiet without being atmospheric in any useful way.