They are different.
A "sip" is a tiny amount. Sipping is considered to be polite and good table manners. If asked to fill in the blank in the phrase "sip of ______ " my first guesses would be "tea," "water," "coffee," "wine," or "sherry."
The word "swig" means a big gulp, and it is usually used in connection with an alcoholic drink. It is considered rude. If someone is taking a swig of something alcoholic, it gives the impression that they are drinking for the effect. They want to get drunk. Asked to fill in the blank of "swig of ______ " and my first guesses would be "rum," "brandy," "whiskey," "beer," or "grog."
Here are some examples of use.
Sip:
"'Good morning, sir,' said Jeeves. He put the good old cup of tea softly on the table by my bed, and I took a refreshing sip. Just right, as usual."--P. G. Wodehouse, "The Inimitable Jeeves"
"The top sergeant was drinking his hot drink in little sips."--John Dos Passos, "Three Soldiers"
"The spy, well used to his business, did not change his unconscious attitude, but drained his little glass of cognac, took a sip of fresh water, and asked for another glass of cognac."--Charles Dickens, "A Tale of Two Cities"
"'I do not quite understand,' returned the uncle, sipping his coffee."--"A Tale of Two Cities"
Swig:
"He looked up, however, at my coming, knocked the neck off the [wine] bottle like a man who had done the same thing often, and took a good swig, with his favourite toast of 'Here's luck!'"--Robert Louis Stevenson, "Treasure Island"
"So I'll meet 'im later on
At the place where 'e is gone --
Where it's always double drill and no canteen;
'E'll be squattin' on the coals
Givin' drink to poor damned souls,
An' I'll get a swig in hell from Gunga Din!"--Rudyard Kipling, "Gunga Din"
"Cut out that baby talk and get me a swig, Teenie. Get me a drink before I get ugly."--a 1920 novel by Fannie Hurst