Does Adele use the same music Bob Dylan wrote?

This is the song Adele plays (http://youtu.be/0put0_a--Ng)

And this is the sheet music (http://www.musicnotes.com/sheetmusic/scorchVPE.asp?ppn=SC0084693)

I’ve tried to play it on the piano, but to me some of the chords don’t sound the same.

Am I just not playing it right? Or does she play some of them differently?

If you leave an answer, please listen to the song and read the sheet music (or even play it). I would be extremely happy if someone could answer my question.

Thanks!

-ben