"Statesman" is complimentary. "Politician" is slightly derogatory.
"Politician" and "politics" often suggest the processes of making deals, compromises, asking for campaign money, and perhaps a trace of corruption or at any rate special favors for cronies.
You could NOT say "politician" here.
"Ladies and gentlemen, it is my great pleasure to introduce to you a devoted public servant, a tireless fighter for the public interest, and the greatest statesman I have ever had the privilege to know: Senator Bumstead Boopfaddle!"
You could NOT say "statesmen" here.
"I see the politicians in Madison [state capital] voted themselves another pay raise yesterday."