The dictionary says both mean "to burn". Obviously there must be some difference, subtle or otherwise. What are they?
First of all note that burn means different things in English:
- Catch fire.
- Be consumed by fire.
- Make something to burn.
Quemar is just burn, in with meaning 3. Meaning 2 will be quemarse or arder.
Prender, however, means to catch fire (meaning 1), put in flame, that is, the act of start burning. It is sometimes used redundantly: prender fuego.
For example, you can say:
La madera mojada no prende bien. - Wet wood doesn't catch fire easily.
But if you say
La madera verde no arde bien. - Green wood doesn't burn easily.
The meaning is subtly different: the former means that it is difficult to start a fire with wet wood; the latter means that wet wood does not give great flames.
But with the following phrase:
¡Los pirómanos quieren prender fuego al mundo! - Pyromaniacs want to burn down the world. ¡Los pirómanos quieren quemar el mundo! - Pyromaniacs want to burn down the world.
The meaning is practially the same.
Note that prender, as is, without fuego following may require some context, because this word has a lot of different meanings.
The meaning of prender as turn on an electrical device sounds regional, from America, I think.