firstly, modern liguistic is DESCRIPTIVE(to describe the way people speak) , whilst traditional grammar is PRESCRIPTIVE(to prescribe the way people speak, or simply, to tell people how to speak and let people know the correct way of their speaking )
secondly, tradition grammar pays more attention to the written form of language, while linguistics attaches more importance to speaking than writing.
thirdly, tradtional grammar has been restricted mainly to SYNTAX, that is, the way of words making patterns to form sentences, while linguistics has a boarder scope for researching, eg. pragmatics, psycholinguistics, sociolinguistics, ect. which, accordingly, are out of the scope of traditional grammar.
of course, there are other differences between the old and new appoaches to language research, such as Diachronic vs Synchronic and so....