At this point in history, there’s no difference. Dry cleaning was once widely known as French cleaning because the process was invented by a Frenchman in the mid-1800s who reputedly noticed that when a kerosene lamp was overturned on a tablecloth, the cloth was rendered remarkably clean. He began experimenting with solvents, and French cleaning was born. The method also became known as dry cleaning, because it doesn’t involve water (and is thus a superior method for cleaning certain fabrics). Today the only difference between a French cleaner and any other cleaner is that a French one may be long established (always a good sign) or have a sense of history (ditto).