This would mean the difference between the project to overthrow the Syrian government through more subtle means (ongoing since the late Bush administration), and a more open intervention that would probably look like Libya a few years earlier.
So, just take a look at Libya. There would be a collapse of state authority, women's rights would be severely curtailed, there would be (even more than have already happened) massacres of minority religious and ethnic groups, and groups like IS would operate pretty openly.
Regionally, the Saudis, Israelis, and Turks would have a freer hand since one of their prime antagonists has been removed. The Iraqi and Lebanese governments would probably have to deal with a surge of IS/Al-Qaeda affiliated groups that would relocate from Syria like similar groups did from Libya following that civil war.
Internationally, since the Russian and Chinese votes on the Security Council would be ignored by an overt US attack on Syria (having actively voted against something like that, rather than merely abstaining as in the Libyan case), I imagine that relations between those countries and the west (supposing that at least a few European countries supported US action, probably the UK and France) would be correspondingly colder. A Syrian intervention in Syria would not take place with the support of the UN, but would be more likely to be a NATO run operation or another 2003-era "Coalition of the Willing".