Palpatine was starting a process to turn the Republic that has stood for thousands of years in to an Empire. Even a Sith as powerful as him could not do it by himself, he needed support from the leaders of the individual worlds to help him maintain control. If he were to start killing them, even one, it might make the others rethink their stance and not support this.
As seen in A New Hope the Imperial Senate (The successor to the Senate of the republic) was still active 19 years after Order 66. That's because he needed to erode the power of the senate away and funnel it in to the Moffs who are loyal to him.
If he for example had executed or outright killed Senator Organa then many other senators would have withdrawn their support and might even align themselves with the separatist...Alternatively if he wanted to do the same to the senators as to the Jedi he would lose control over the chain of command and member worlds might just go up in Rebellion . This would mean that Palpatine would be forced to conquer every world one by one. And seeing that even with the full support of the Senate it took the Empire 3 years to clear out the last Separatist holdouts... now imagine if half the republic switched sides to the separatists.
And lastly, the senators were no real threat to his power unless they banded together, something that at the time of the formation of the Empire was no reason to.