"Periódico" seems more common to me. "Diario" can also mean "diary", or "a diario", daily.
It is typical by country I understand. I may have it reversed, but I think in Argentina they call the newspaper diario while in Mexico they use periódico.
I called it a magazine and got marked wrong. I have heard a paper called "el jornal" (phonetic) by native speakers in Arizona, but haven't seen it in the dictionary. Seems newspapers and magazines is a topic subject to lots of regional variety.