I think syntax is concerning with the theories of syntax like structuralism, behaviorism, traditional, and informational since each school has it's own rules and theories while Grammar is in regard to the rules that form the correct text via applying or depending on the school.

Is this a common understanding of these terms?

  • If you want them to have a simple meaning, syntax is about Part-Of-Speech, and grammar is about sequencing them into phrases and clauses.
    – amI
    Mar 20 '18 at 23:30

I assume that was supposed to be a question, like "is that how linguists see it?". Syntax is not about theory of syntax: theories of syntax are about syntax. Syntax is generally understood to refer to the rules of how sentences and clauses are put together, and a theory of syntax is a theory of those rules. Some theories may include large chunks of "morphology" within the scope of syntax, others may include some aspects of meaning. Some theories actually exclude word order from the scope of syntax. There are a lot of school-dependent differences. I don't think there is any theory that say everything about language structure is "syntax", so usually there is something like "phonetics" or "phonology" (or both), "lexicon", and "semantics".

"Grammar" as used by linguists refers to pretty much all of the structural facts of a language, encompassing semantics, phonetics, phonology. This is in contrast to popular definitions of "grammar", which refers to "proper grammar", that is, normative rules about how you should speak in a context. Sometimes there are official academies that declare what is proper, otherwise (and especially in the case of English) is is via subjective apostolic tradition, where one may adjudicate a matter by reference to The Chicago Manual of Style (or some other text).

Linguists may study normative grammar, but typically without approving of the underlying dictates. For example there is a linguistic study, A Dictionary of English Normative Grammar which studies what normative grammarians have said. Linguistics is a science, meaning that is describes what the facts are, and does not make claims as to how people should behave.

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