As well, often, when people talk about grammar, they’re talking about morphology (mostly inflectional), ie changes on the endings of verbs (conjugation) and nouns (declension), etc. Or, if not morphology, they’re talking about rules of syntax, ie sentence construction - where does the verb go, etc.
This is probably why you probably get people saying things like “Chinese has no grammar”. On the one hand it’s kind of a nonsensical statement, given the definition of grammar I gave above, but I understand where people are coming from. The words in Chinese don’t change very much like they do in Latin, so it doesn’t have much “grammar”.
At any rate, the larger difference is the approach to grammar. Traditionally, grammar was seen like a schematic of the language, and students tried to memorize the schematic, or understand it academically, in the hope that they would they be able to use all these rules to produce correct language. We now know that this doesn’t happen. We learn from content. And some morphological (word-level) or syntactic (sentence-level) features take some time to sink in, just as semantic (meaning) features do.
But explanations of sentence construction, or changes in word ending, or the semantic content of a word (ie its meaning as found in a dictionary) can be very helpful when you want to understand something. Trying to learn “grammar” in advance is like trying to learn “the dictionary” in advance. It doesn’t work like that.
What grammatical and lexical information is helpful for learners, and at what stage in the learning process, is an interesting and difficult question. And it’s probably different from person to person due to their studying habits, interests, and personalities.
Unlike some, I don’t mind drills and exercises. I wouldn’t do them exclusively, but as a small part of an overall program. I consider my Anki flashcard review to be a kind of drilling.