Would not a vote of "no" send a message to Assad that the US and it looks like the majority of the world doesn't care what he does in Syria as long as he doesn't directly attack them? Would he then not be more apt to use any weapon he so desired if a situation arose that he thought could only be solved by using WMD? Basically his thoughts could be "I did it once and nothing happened so I'll do it again". By the way I am still undecided, just playing the devils advocate.
To begin with, I do not think it is clear that Assad authorized the chemical strike. It is possible that it was a rogue element in his administration, or that some other party did it.
But assuming that Assad did indeed use the chemical weapons, yes, he will likely feel like he got away with something. But I doubt he'd think this would give him free-reign to act however he wants, seeing as this act brought him very close to destruction, and that another strike might serve to be a tipping point.
As for making America appear weak, my response is that I could care less. America's military abilities are astronomical, and they'd have to be blind to think that an instance of restraint somehow magically means that next time we can't blow them to bits. Furthermore, I'd hope that restraint is our general policy in the future. America shouldn't unilaterally be the world's police, and if this is America's first step in abdicating that role, then the sooner the rest of the world wakes up to that fact, the better.