For me drizzling is more continuous an a bit heavier than spitting. You might say it's been drizzling all day, this is unlikely for spitting.

I'd agree that they are synonyms if we are talking about rain. Or at least there's enough flexibility about the exact meaning that most people would use them as synonyms. They both mean 'light rain', and I don't think people could define the exact amount of light rain involved. As suggested in another answer "spitting" is British English. And "spitting" has a completely different meaning in other forms of English. It means deliberately ejecting saliva from the mouth. So, if the context is something different (not rain) then the two words are not synonyms. For example to describe pouring a small amount of liquid over a cake then you would use "drizzling" and definitely not "spitting"! So which is the correct answer? Well, if you are communicating in international English (not BrEng) there is a risk that people will misunderstand, and so I guess it is possible to argue that "drizzling" is the better choice.

I use "it's drizzling" often. I never say "It's spitting." Perhaps that is a British expression. удачи!