The thing is when President Obama planned to bomb Syria, it was going to be a "shot across the bow" which would punish Syria and hopefully deter Assad from using chem. weapons again. The goal of the proposed bombing campaign "wasn't to change the situation on the ground" or to initiate regime change.
So the most likely result for the bombing campaign is for the US to destroy some of the chemical weapons, and ultimately failing to prevent future chemical massacres. Ultimately, Assad would be strengthened for "fending off the United States." The result would be no different than Reagan's airstrikes on Libya or Clinton's on Iraq.
That comes down to the real question are they strikes to say you are doing something because your red line was crossed or is this Obama at least downing Assad's air power and putting up an air power containment of the ISIS situation so it doesn't pour into Iraq. Obama really had four options... do nothing. Do a few half ass strikes and declare victory. Put up a containment strategy and hurt Assad's ability to bomb civilians and ISIS to cross on mass into Iraq. And, finally a full on air power against Assad and ISIS to wipe them out. The final one was never really in the cards given it might require boots on the ground when the regime implodes and by that point probably would. The other three were politically viable potential options.
If this is a few days of bombing while avoiding Assad's air power and then saying Mission Accomplished then nothing changes other then the WH has a bit more credibility when Putin is thinking on invading Ukraine.
A full on containment strategy using air power and the return of more then 100 advisers to Iraq (the number we sent in mid 2013) being put up though when the ISIS and Assad problem was contained to Syria would have huge ramifications though most of them good for the US.
America will be leery about the whole thing politically until bombs start dropping on not just Assad, but IS as well which ups the date American heads start coming off and then again suddenly the public will be 2/1 in favor of it.