Syria's population is 74% Sunni muslim, 16% other muslim sects and 10% Christian. The ruling elite are from the minority Alawi muslim sect (12%) who make up most of the socialist/secularist Ba'athist gov't.
The Syrian muslim brotherhood, who regard Alawi's as borderline heretics, have engaged in armed uprisings before based on the city of Hama. In 1982 the gov't used heavy artillery, levelling parts of the city and causing thousands of deaths, to crush the fundamentalist opposition which has been relatively quiescent until spring 2011.
When Asad's father died in 2000, they had to immediately ammend the constitution to allow his son Bashar to take the presidency: the constitution previously mandated 40yrs as the minimum allowed - Bashar was only 34. The referendum in 2007 confirmed Bashar al-Asad in power for a futher 7-year term.
So, is the west saying the Asad administration is illegitimate because it's basically an authoritarian, totalist regime masquerading as a democracy? Sure people vote there, but the votes are rigged and the choice is limited. Doesn't that remind you of somewhere else?
So the accusation of 'illegitimacy' seems to be based on the observation that the government is, in reality, a stitched-up, totalitarian state and the claim that it is a free multi-party democracy a mere projection, a hologram to present to the outside world which fools nobody.
The command and control of the military and the internal security services is all in Alawi control, but many of the conscripts, (compulsory since late 2011), to the army are Sunnis - so defections from the ranks is likely to grow as they will refuse to kill their Sunni brothers.
But the insurgency seems unlikely to succeed against a vast, well armed regime and ruthless internal security forces, unless they get the active backing of external forces which Russia and China will veto. Russia's interest is presumably to keep a toe-hold in Arabia, maintain it's naval base in the med on Syria's coast, and keep a valuable customer for its arms sales. China's interest is, I suppose, to protect its now billion dollar investments in the country. It's not clear to me that their support for Syria will not badly backfire on them later on so this may be quite a big gamble for both.
Maybe Asad could've headed this off by engaging in some meaningful reforms much much earlier, ones that would have compromised sufficiently to make the insurgents back off yet still leave him in overall control. Who knows?
Where is the Arab League in all this? What happens if the Asad regime falls and a fundamentalist Islamic theocracy is the outcome? What will Israel do then? How far is Turkey going to take its opposition to the Asad regime?
What a mess!
[EDIT] Some further reading: The Long Road To Damascus