A Spanish person is someone who is a native Spanish speaker. In most cases that only applies to people from Spain, but some people will also say a Mexican is Spanish. A Spaniard is a citizen of the Kingdom of Spain and includes Basques, Galicians, Catalans, Asturians, and even Arabs, Chinese, or Russians if they have a Spanish passport.

When "Spanish" is used as a noun, it means the "The Spanish language". It cannot be use to mean a person from Spain. Do you speak Spanish? (good) Are you a Spanish? (incorrect) When "Spanish" is used as an adjective, it can refer to anything from Spain. I like Spanish architecture. Spanish food is my favorite. I can speak the Spanish language. (I can speak Spanish.) If you want to refer to a person from Spain, you can also use the word "Spaniard". My friend is a Spaniard. My friend is a person from Spain. My friend is a Spanish person. (this is grammatically correct, but very awkward) You only use the word "Spaniard" to refer to people from Spain. People from Mexico are Mexicans, not Spaniards. People from Peruvians. Etc. But you can use the word "Spanish" as a noun (the language that is spoken in Spain, Mexico, Peru, etc.) or as an adjective to describe other things that have been brought from Spain to other countries.

"Spanish" is an adjective, denoting that a person (or object) originates from Spain while "spaniard" is a noun, meaning "person from Spain". Do people actually use "spanish" to refer to people from Mexico or Latin America?

The difference is an important grammatical one. When referring to a person, the word 'Spanish' - like all other nationalities ending in 'ish' - is an ADJECTIVE only. You cannot say, for example, 'He's a Spanish'. Spaniard is a NOUN. You can say: He's Spanish . (adjective) He's a Spanish man. (adjective + noun) He's a Spaniard. (noun) You are also correct in thinking that the noun 'Spaniard' is a little old-fashioned and not used so much these days.

I heard it many time. Specifically, from USA citizens and I've got the impression that they refer to Spanish people not from Spain.