Well, obviously Germany wouldn't need to keep diverting forces and commanders to Italy and points south to bail them out. OTOH, the WAllies will need to put far fewer forces into defending the area (it'll be mostly a nice Allied lake), and there's a non-trivial chance of a FFO scenario. If Mussolini stays out of the Balkans, then that entire area probably stays fairly placid (at least the coastal areas).
Reasonable chance given the Allied control over the area of making Dragoon the main Allied invasion of France--it fully secures Allied supply lines and has certain advantages over a cross-channel invasion, especially in terms of defenses. The difficulty in air cover, though, means that the Allies are going to need Spanish (ha!) or Italian (ha!) support, or will have to invade Corsica first so do CAPs over the region. But considering how successful Dragoon was IOTL, such an invasion might end up being more successful in certain terms than our Overlord.
Overall, this leads to a bit more German success on the Eastern Front, but conversely an easier time for the WAllies and possibly an earlier and differently located invasion. The invasion will have some more forces available to fight it, but will probably be stronger than Overlord IOTL, which will mostly cancel each other out. Germany might end up defeated a few months earlier, and the WAllies might be able to penetrate deeper.
@Commissar: The Italians are not noted for either having great quantities of oil or being located on the Pacific. I find the likelihood of Japan getting oil and other raw materials from them to be...low.