Yes, 'I got you under my skin' is a great song from way back when.

It's not entirely grammatical, mind you.  The two 'correct' ways of saying this are:

'I've got' 

or

'I have'.

The first is more common in spoken and informal language, the second in written and more formal language. Some people may tell you that 'I've got' is British and 'I have' is American, but in fact it's not that simple.

Just remember that 'I got' is ok for song lyrics and very informal use only.

The only difference that I can think of immediately, is that the phrase "I've got you under my skin" has been famous for a long time since it was written as the title of a song by Cole Porter in 1936, and made even more famous by the voice of Frank Sinatra decades later.

I can't sing and swing to "I HAVE you under my skin"...

The correct is: "I've got..."

There's no difference. They mean exactly the same thing.