The biggest difference is the magazine, but there are many. I believe that the SAN classics/specials are much superior firearms.
The 556 is a commercial rifle and is much cheesier than the military bred SANs.
The 556 lowers are machines aluminum compared to stamped steel on the 550 series. They uppers are of the same design. Also, I do not believe the 556 is brazed and welded, only welded.
Accuracy of the 16" barreled 556 was not as good as the 9" barreled SG552 @ 100 meters. Overall fit and finish was not as good. No nightsights. As I said, I really like the 556, solid, reliable good gun. The only thing that really bothered me was the accuracy, just not on par with the Swiss Arms products. In the end no-one would trade their SG552 for a Sig Sauer 556.
A buddy of mine works at Swissarms, they received a SIG 556 for trials. They put it through the same tests as the 550 has to pass.
According to him it did surprisingly well, but still failed in the end because it's just not as durable as the original.