But / But still coffee with sugarStating contrast or contrary outcome

Contrast vs. Contrary Outcome

CONTRAST	CONTRARY OUTCOME
But is a connector that introduces a clause with an idea, opinion or action that is different from the other clause.  It indicates a contrast.	But still is a connector that introduces a clause with an idea, opinion or action that is contrary to the idea, opinion or action in the other clause.  The outcome is contrary (i.e., illogical, irrational) to expectation.
BUT / HOWEVER He asked for another cup of coffee, but there was no more. He asked for yet another cup of coffee. But there was no more!  †(stronger contrast) He asks for some and finds out there is no more.	BUT STILL / THOUGH There was no more coffee, but still he asked for another cup. He knows there is no more coffee and asks for more, (Perhaps, he hopes someone will  buy or make some more.)
He asked for another cup of coffee. However, there was no more. He asked for another cup of coffee; however, there was no more.	Though there was no more coffee, he asked for another cup.. He asked for another cup of coffee though there was no more.

Also see but / toguht for implied meanings
† but beginning a sentence  

Connectors for Contrast

INITIAL			MEDIAL
Each of these connectors states a direct contrast and can be placed before either clause without a change in meaning.   A comma is used to separate the two clauses.			Each of these connectors states a direct contrast and can be placed between the two clauses.  A comma separates the two clauses (except for however, a transition word, and in contrast to, a preposition.)
CONNECTOR	CONTRAST	STATEMENT	STATEMENT	CONNECTOR	CONTRAST
While (conj)	I like sugar in my coffee,	Edward likes black coffee.	I like sugar in my coffee,	while(conj)	Edward likes black coffee.
Whereas(conj)	I like sugar in my coffee,	Edward likes black coffee.	I like sugar in my coffee,	whereas(conj)	Edward likes black coffee.
On the one hand, (trans)	I like sugar in my coffee.		I like sugar in my coffee,	but (conj)	Edward likes black coffee.
On the other hand, (trans)		Edward likes black coffee.	I like sugar in my coffee	in contrast to (prep)	Edward who likes black coffee.
			I like sugar in my coffee.	TRANSITION WORD comma However, (trans)	Edward likes black coffee.
			I like sugar in my coffee.	In contrast, (trans)	Edward likes black coffee.

black coffee – has no sugar
On the one hand,  and On the other hand, is a paired expression.
 Also see comparisen

Connectors for Contrary Outcome

CONNECTORS BEFORE SITUATION CLAUSE			CONNECTORS BEFORE CONTRARY CLAUSE
Each of these connectors occurs with a pair of independent clauses.  It introduces the situation-clause.			Each of these connectors occurs between a pair of independent clauses and introduces the clause with contrary outcome (idea, opinion or action).
CONNNECTOR	SITUATION	CONTRARY OUTCOME	SITUATION	CONNNECTOR	CONTRARY OUTCOME
Even though (conj)	I shouldn't eat sugar,	I do (eat sugar).	I shouldn't eat sugar,	but still	I do.
Though (conj)	I shouldn't eat sugar,	I do.	I shouldn't eat sugar,	yet (conj)	I do.
Although (conj)	I shouldn't eat sugar,	I do.	I shouldn't eat sugar,	but (conj)	I do anyway.
ADVERBIAL PREP + CLAUSE Despite the fact (that)	I shouldn't eat sugar,	I do.	I shouldn't eat sugar.	ADVERBIAL PHRASE comma Even so, (adv)	I do.
In spite of the fact (that)	I shouldn't eat sugar,	I do.	I shouldn't eat sugar.	In spite of that,	I do.
Regardless of the fact (that)	I shouldn't eat sugar,	I do.	I shouldn't eat sugar.	Nevertheless, (adv)	I do.

contrary (adj) – ideas, opinions, or actions that are completely different and opposed to each other; one is illogical, unexpected or unlikely with the other
Also see:   because thoguht | adverbial preposition

Punctuation

INITIAL-SENTENCE POSITION			MID-SENTENCE POSITION
Each of these connectors introduces a situation clause that is followed by a comma and clause stating an unexpected, surprising or illogical outcome.			Each of these connectors can be placed mid-sentence before the situation clause. No comma is used.
CONNECTOR	SITUATION	CONTRARY OUTCOME	CONTRARY OUTCOME	CONNECTOR	SITUATION
Even though	I shouldn't eat sugar,	I do (eat sugar).	I eat sugar	even though	I shouldn't.
Though	I shouldn't eat sugar,	I do.	I eat sugar	though	I shouldn't.
Although	I shouldn't eat sugar,	I do.	I eat sugar	although	I shouldn't.