From here I know "put it" means "say it, express it". But have them any distinction of usage? When to use "put it" and when to use "say it"?
To put it can be used in place of to explain it; it means to express (a thought or comment) in a particular way, form, or language.
-Let me put it this way: it's only a funny story if you know John. (explain)
-When you put it like that, I can understand it better. (express)
To say something means you are using words to express a thought; it could be anything, including an explanation. You can use say it for put it, but not usually the other way around.
-Say it more slowly, please.
-What did she say?