Still and Yet

They are not the same.

Yet can be used in the following situations:

With present perfect in the negative:

         “I haven’t finished eating yet.” > “I am still eating.”

It is also used in questions, 

         “Have you finished your chores yet?”

With ‘to be’ in the affirmative simple present:

“I am yet to buy a gift for my girlfriend.”

With ‘to have’ in the simple present:

“I have yet to study for tomorrow’s test.”

Yet can also replace the conjunction ‘but’:

         “I lowered the price, but he didn’t accept it.”

         “I lowered the price, yet he didn’t accept it.”

Still is used to mean that a situation is the same as before:

“I am still at work.” > “I haven’t left work yet.”

“After 30 years, he still drives without a seatbelt.”

With the present/past continuous for the same effect:

“I am still writing the email.” > “I haven’t finished yet.”

“When we left, he was still talking about how he got promoted.” > “He hadn’t finished yet.” 

Still can be used with the present perfect in the negative:

“I still haven’t finished my lunch.” > “I expected to have already finished.”

“She still hasn’t paid me.” > “I expected her to have already paid.”

Still is used instead of ‘even so’:

“Samantha loves sweets. Even so, she avoided eating dessert at the party.”

“Samantha loves sweets. Still, she avoided eating dessert at the party.”