What are the major conflicts between traditional and modern approaches to the study of language?
There is no one system of "traditional" or "modern" linguistics, but rather a series of controversies or differences in approach, many of which have continued over several centuries.
The study of linguistics has changed significantly since its Graeco-Roman origins. The first radical change was the identification of “parts of speech” by the Greek sophists. The next major change was the analogist-anomalist controversy, in which one side argued that language began with completely regular forms and irregularities were errors that developed gradually, with the other arguing the reverse. Christianity brought the theory of Adamic naming and the notion that the goal of linguistic study, inter alia, was to recover the Adamic language -- a movement leading, mutatis mutandis, to the investigation of proto-Indo-European. One controversy starting in the 18th century and still continuing is whether linguistics should be descriptive or prescriptive. A 19th century conflict, again still active, was whether language should be studied synchronically or diachronically. Finally, in the 20th century, due in part to recording technology, there was a shift from language studied as primarily written to a focus on the oral.