I'd say an appliance has a very specific function. Wash clothes, make coffee, cooking, keeping things cold, and is usually named after that function. Washing machine, coffee maker, refrigerator. "Stove" would be an exception; they used to be employed for more than cooking, and the name carried over (they were also used as "heaters", keeping a home warm). Likewise an "air conditioner" is an appliance, meant for environmental cooling, but it is difficult to find a name that doesn't imply a refrigerator (even though that is what it is).
Appliance are single purpose, and widely used and understood. Often an appliance will have a very limited job time, they are started and stopped (or stop automatically). Like a coffee maker, or ice cream maker, or washing machine. In a way that is even true of air conditioners and heaters and dehumidifiers, they are just started and stopped automatically, to "correct" the temperature in an environmental space. Similarly for a refrigerator.
Usually, an appliance delivers some kind of convenience, it does a job for you.
A "device" is a more general term, I think all appliances are devices, but the terms are not synonymous. Many devices can be for singular purposes (like a forklift, or ditch digger, or a manufacturer's machine to fill bottles) but we don't call them an "appliance" because we don't expect many people to have them, and don't think of their "job" as a ubiquitous purpose consumers might want done.
Or, it is possible that whatever the device does is not commonly even thought of as a job to be done: A TV is a device, few of us would call it a video appliance. Or call our stereo a music appliance. Likewise a modern phone, or computer (same thing). The things these devices do are not commonly thought of as "jobs" to be done. They aren't seen as providing a convenience.
The universe of "devices" envelops the universe of "appliances" but the line between them can be ambiguous.