To me, Arts & Crafts is the "true" version, which came over from England in the late 19th century. They were inspired by Japanese joinery when the empire opened up to Westerners. "Arts & Crafts" style in America ran into the heavy Catholic mission furniture on the West Coast, which used to be part of Mexico. The result was "Mission" style, which you'll see in CA wineries, etc.
I believe "Craftsman" was a production/marketing twist on true "Arts & Crafts" handmade furniture that was sold in catalogs (if you Wikipedia.com these terms you'll get some interesting backstories). You can also apply "Craftsman" to bungalows (smallish houses with exposed architectural details like rafter tails, etc.).
Greene & Greene was a response to the austere utilitarianism of Arts & Crafts, so added decorative details like cloud lifts and celebrated their joinery instead of trying to minimize it by making breadboard ends exaggerated and plugging all screw holes with ebony square plugs.
You have to be careful with Stickley because Gustav and L & JG made similar looking furniture, but the production quality is very different. Don't forget giants like Morris, Limbert, Hall Brothers, etc.
Don't even get me started on "Prairie" style, which has a definite Northern California/Pacific Northwest influence. Basically, unless it's specifically a Greene & Greene piece (or some other distinct designer), it can probably be labeled "Arts & Crafts", unless it looks really heavy and should be in a monastery so it'll be "Mission". If it was hastily put together in a factory, I'd call it "Craftsman". I don't know if there's any connection with "Craftsman" style and Sears' "Craftsman" brand, but I think you could buy an entire house/bungalow from the catalog in the 50's.