Rabbi Aaron Rakeffet suggests the following distinctions between Rabbi Samson Raphael Hirsch's Torah Im Derech Eretz and Rabbi Joseph Dov Soloveichik's Torah UMada.
Rabbi Schachter's view, as I understand it from hearing a few of his lectures, could be called a practical-minded intersection of the two. He appears to have a mixed view on the theoretical value of fine arts ("we got tickets to the Symphony, it was a German composer. Everyone else said it was so inspiring, I could see the Nazis marching, I could feel the wickedness!") But he understands in a very real way that people need decent educations, if nothing else to earn a living, as well as to understand reality so that halacha can be applied to it. He will frequently cite general history as a source of lessons. On the other hand he sees little value in a dual curriculum requiring studies such as the New Testament (other than perhaps a history course which may occasionally cite a line or two to provide historical context). When challenged by the precedent of such a curriculum, he admitted "I say it was wrong then, and it's wrong now."