What is the difference between a periódico and a diario? If both mean exactly the same thing, which one is preferred in regular colloquial usage in Mexico vs. other Spanish-speaking countries?
Periódico is the common word to say in Spain.
It has some sinonyms, including diario:
Mainly are the same, but Diario is a Periódico published daily.
diario is the common word -by far- in Argentina.
periódico is understood here as a formal (or "foreign") synonym.
Stricly speaking, "diario" is more restricted in meaning (daily newspaper), but common usage does not respect that much anymore - one could even hear about a "diario semanal".
I have always used the word "periódico" for newspaper.
Actually in some cases it can be used as just simply "paper" (not as in plain regular paper that you write with).
Diario can mean "every day" (which I am certain is not what you mean), "daily", or "diary".
If the conversation is about some text and I hear the word, "diaro", I think of a "diary". I personally do not use the diaro to describe a "newspaper".
I hope this helps.