When I was in college ('99-'03), phonics was out and the holistic approach to teaching reading was in. Since then, I think the pendulum has swung back toward phonics again.
Anyway, the "traditional grammar" approach is a lot like teaching phonics. It is teaching grammar by isolating parts, rather than looking at how grammar works inside of sentences and paragraphs (and conversations). This type of teaching is inadequate when it is never brought back to looking at the whole. There needs to be a balance between "traditional" grammar lessons and big picture "why is this important" lessons.
Traditional grammar by itself is like teaching vocabulary lists without any context. Students may be able to learn rules and regurgitate answers for a quiz or test, but ultimately, without application (or a reason to care), they lose it.
I've also found in teaching writing that students are often able to pick out major grammatical mistakes without knowing exactly how to fix them, or worse, they cease to catch the major mistakes. Again, traditional grammar (isolated focused lessons) makes it hard to see multiple problems going on at the same time. However, without the traditional grammar lessons, problems are identified simply by "it sounds wrong" rather than by an actual understanding of how the language works.
The traditional grammar approach might be inadequate in studying a language when the activities that are offered consist on repetition, cloze and rote memory activities, and mere reading and writing without context.
For example, if one is going to teach animals, and all you do is repeat each name, give it a meaning, fill in blanks, or use the words in simple sentences, there will never be an opportunity to actually get to know the meaning in context. It would be very different that dialogue and responses are required so that an interaction can take place in which the target words and many more can be learned in one activity.
The new approaches show that interactive reading, dialogues, constant feedback, and relevant, contextual activities are the best way to acquire a language effectively.