MEANING

We use STILL to say that something has not finished  (we look to the past)
- It's still raining  (it started raining in the past and it has not stopped)

We use YET to say that we expect something to happen and it has not happened... but we think it will.  (we look to the future)
- Annie hasn't arrived yet. She's late


WORD ORDER

STILL goes in mid-position (before normal verbs and after special verbs: modals and auxiliaries)
- I still love you
- I can't still see anything
But in negative sentences it often goes before all verbs, normal or special!, especially when verb+NOT are contracted.
- I still can't see anything

YET always goes at the end of the sentence
- The movie hasn't started yet


USAGE

We use STILL in all kinds of sentences

(+) I didn't forget. I'm still thinking about it
(-) I've read your explanations, but I still can't understand
(?) Are you still working for Microsoft?

We only use YET in negatives and questions

(-) Wait for me, I haven't finished yet!
(?) Has your father arrived yet?

STILL and YET in negatives and questions

Remember that STILL looks to the past and YET looks to the future

1- I still don't have a job  (I've been looking for a job in the past but I don't have a job)
2- I don't have a job yet  (but I will, sooner or later)

Case 1 thinks of the situation as something that was negative and is negative.
Case 2 thinks of the situation as something that is negative but will be positive.
So the situation is the same, but in 1 we are pessimistic about it and in 2 we are more optimistic.

In questions we also have the past/future look: STILL asks if a past action continues or not, and YET asks if something is true or we must wait a bit more.

- Is it still raining? (the past rain continues or not?)
- Is it raining yet?  (is the action true now? if not, I expect it to be true in the future)

AS CONNECTORS

STILL and YET can also be used as connectors to express contrast. In that case they both mean exactly the same:

- I believe you. Yet, I think you are exaggerating = I believe you. Still, I think you are exaggerating

(click here for the contrast connectors)

Note for Spanish Speakers

AFIRMATIVAS: STILL se puede traducir por: Todavía, Aún, o también por: Sigue + gerundio
It's still raining= Todavía está lloviendo = Aún llueve = Sigue lloviendo

NEGATIVAS:  STILL= Sigue sin       YET= Todavía/aún:
- I still don't have a job = sigo sin tener trabajo (¡qué vida esta!)
- I don't have a job yet = todavía no tengo trabajo (pero estoy buscando, a ver si pronto me sale algo)

INTERROGATIVAS: STILL= Sigue?       YET= Ya:
- Is it still raining? = ¿Sigue lloviendo?  (¿aún no ha dejado de llover?)
- Is it raining yet? = ¿Ya está lloviendo? (¿Ya ha empezado a llover?)