The interesting thing about the Axis of Awesome four chords is that they are the same four chords that were in the stereotypical 1950s/early 60s pop song (Heart and Soul, Teenager in Love, Up on the Roof) but the order has changed. In the key of C the 50s version was C, A minor, F, G (or sometimes G7 instead), and in the Axis of Awesome / Adele version it's C, G, A minor, F. Of course Axis of Awesome cheats a little bit because they play the 4-chord parts of those 30-odd songs (though in fairness those tend to be the best known parts).
Yes, and this is the crucial difference that make new songs sound "modern". The Beatles were one of the first to use the 1,5,6,4 sequence (Let it be). Since then the 1,6,4,5 sequence has consistently lost ground, while 1,5,6,4 sounds modern. The chords are still the same.