Sink simply means that something, perhaps you, go down in something. You can sink to the bottom of a lake, and then come back up. This is not forever.
Drown means that you die because you get too much liquid of some kind in your lungs. Normally, this means you sink in water, try to inhale it, and die that way.
Also, if something is not alive or doesn't breathe air, it cannot drown, but it can sink. Fish and ships can both sink, but they can't drown.
Drown can also mean "to be covered, enveloped, or overwhelmed by something". I could say "He is going to drown in his work." I do not mean he is going to die, but simply that he will be overwhelmed with work.
Sink can mean almost the same thing, except you do it to yourself, and it is not a bad thing. "I am going to sink into my work." I mean that I will surround myself with my work and ignore other things.