One of my teachers, Rabbi Dr. Yosef Tabory, encapsulates the difference between academic and traditionalist forms of study with a single word: Context.
Traditionalist Torah study analyzes the words of Torah scholars over the ages without regard to external factors aside from considering whether they were Rishonim or Acharonim. (Note: An exception seems to be made for the Moreh, which is often disregarded on the grounds that it was written "for kiruv" etc.)
Academic study analyzes the words of Torah scholars over the ages with the aid of examining the context in which they were written. What societal, cultural, intellectual, political factors could have been involved, if any?
The traditionalist approach regards the latter as near-heretical. There is a letter from R. Dovid Cohen in an old issue of The Journal of Halachah where he makes this point (I don't have it with me, if anyone can send it to me I will include it in this post.)
This difference lies behind some (but not all) aspects of the dispute over Rashi's view of corporeality. It is also the reason why some people objected to my "Jumping Elephant" essay. And it is also the reason why some people will be objecting to my forthcoming essay on comparing the development of the shiur of kezayis in Sefarad and Ashkenaz - in which a significant factor is that there were no olives in Ashkenaz.
Note that I am not passing a value judgment on these different approaches. If we are talking about reaching historical truth, then I consider the academic method far superior. But I am not judging which approach is more valuable from other perspectives, such as for the mitzvah of Talmud Torah or for reaching psak. There is a fascinating exchange in the current issue of Hakirah on topic of whether calling a form of study non-historical means that it lacks value.
There are other significant differences between traditionalist and academic modes of study, but the concept of context is very significant.