grammatical vs grammatically correct

To say that a sentence is grammatical is to say that it conforms to the rules of English grammar as found in the way in which native speakers normally use the language and, in the case of Standard English, as codified in various academic works of grammar such as ‘The Longman Grammar of Spoken and Written English’ and ‘The Cambridge Grammar of the English Language’.

The introduction of 'correctness' introduces value judgements which tend to reflect the prejudices of the writer rather than the way the language is actually used. Some people, for example, will say that it is incorrect to place an adverb between to and the plain form of the verb, or to use less when describing things that can be counted. Those are matters of opinion, not of fact. Describing any construction as incorrect is unhelpful and inadequate. That is why, in most cases, it it makes more sense simply to say whether or not a construction is grammatical.

Thanks, centrox, for the detailed and clear reply.

Grammatical = adjective
Grammatically = adverb