In linguistics, the term usage relates to the habits of language use among a language’s native speakers, particularly with regard to the meanings of words and phrases. Grammar relates to the functions of words, the construction of sentences, and how words combine to make sentences. Usage evolves continually, often rapidly, and is subjective in many ways. Grammar is more rule-based and objective and tends to evolve more slowly. The two subjects are often lumped together, usually under the term grammar, but for people who study and teach about them, they warrant very different approaches.
Examples
Usage
Usage of the word became more complex with the dawn of the gay-rights movement. [NPR]
I explained that “gotten” is a Renaissance usage found throughout Shakespeare; he uses “ungotten” too. [Guardian]
But the word I’m looking for, apparently, is downsize, a nasty bit of jargon which has crept into ordinary usage. [Canberra Times]
Grammar
This charge of “word abuse” brings to mind the debate about when the rules of grammar should bend to meet the demands of the “mob.” [National Post]
Note that even then, however, he preserved the lack of grammar that is central to Dylan’s best work. [Irish Times]
Grammar books indicate quotation marks qualify the meaning of a term and thus Ryan — in Ayn Rand fashion — disputes that health care is a right. [Washington Post]