I'm thinking of the way the terms are applied if you walk into a department store here in the U.S.
Appliances are devices that perform work of some kind for you - a dishwasher, a toaster, a washing machine or dryer (for clothes), a vacuum cleaner, a refrigerator - while electronics are primarily entertainment devices based on electronic technology - televisions, stereo systems, Ipods, videogame consoles.
Computers are electronic devices but usually have a separate section in our stores here in the U.S.
These are the everyday uses of the terms here as sections in a store. There are other meanings for both words. A prosthetic arm, for example, can also be called an appliance. Anything electronic - a guidance system for a missile, for example - can be included in electronics. I think you're looking for the way the words are used by retailers and consumers, though. I may be wrong.
Just to give some more tangible examples: Appliances include such things as: stoves( ovens), fridges, washing machines, dryers,dishwashers, microwaves, and even small kitchen appliances such as a a food processor, blender, coffee-maker etc. Most electrical tools which pertain to kitchen activities are called appliances.
Electronics are devices used mostly for entertainment such as a television, stereo system, computer, portable devices including ipods, blackberrys, digital cameras and cell phones. Also, a telephone, fax machine, printer/photocopier would be known as electronics.
Perhaps a way to categorize would be where they are used. Appliances are found mostly in the kitchen or laundry room while electronics are found in office settings, and living rooms/family rooms or are items that you carry around with you. You cannot easily transport appliances. Hope this helps.