What's the difference between 끄다 & 꺼뜨리다?

끄다 means "put out a fire" or "extinguish." Here are some examples:

• 불을 밟아서 끄다 -- stamp out a fire
• 촛불을 불어 끄다 -- blow out a candle
• 물을 끼얹어 불을 끄다 -- put out a fire with water
• 담요로 덮어 불을 끄다 -- smother a fire with a blanket
• 두들겨서 불을 끄다 -- beat out a fire

끄다 can also mean to "switch off (a light or an electrical appliance)" or "turn off (an engine)"

• 불을 끄다 -- turn off the light
• 텔레비전을 끄다 -- turn off the TV
• 시동을 끄다 -- stop [kill] the engine

When you say 불을 끄다 (put out the fire), you are saying that a fire or light is put out or turned off intentionally, but when you say 불을 꺼뜨리다, you are saying that a fire goes out by mistake. Here is the definition of 꺼뜨리다.

꺼뜨리다 -- put out a fire [light] by mistake; let the fire [candle light] go out (by mistake)

• 불을 꺼뜨리지 않다 -- keep the fire from going out
• 불씨를 꺼뜨려 버렸다 -- Now there is no live coal to start the fire.

Since 꺼뜨리다 implies a mistake, it is usually used in the negative form to warn against making that mistake, so 꺼뜨리지 마세요 (Don't let the fire go out) makes sense, but 꺼뜨리세요 (Let the fire go out by accident) does not. Therefore, if you want to tell someone to put out a fire, you would say "불을 꺼주세요," not "불을 꺼뜨리세요."

Here are the sentences to remember:

• 불을 꺼주세요 -- Put the fire out (intentionally).
• 불을 끄지 마세요 -- Don't put the fire out.
• 불을 꺼뜨리지 마세요 -- Don't let the fire go out (by accident).
• 불을 꺼뜨리지 않다 -- [I] won't let the fire go out.

The difference between 꺼뜨리지 않다 and 꺼뜨리지 말다, I assume, is that you use 꺼뜨리지 않다 when you are promising that you or someone else will not let the fire go out, and you use 꺼뜨리지 말다 when you are telling someone not to let the fire go out. 말다 is used for commands, and you would not give yourself a command.

Also, for me, 꺼뜨리다 is somewhat confusing because of such words as 떨어뜨리다, which means "to drop (something)," either intentionally or unintentionally.