The above comments and answer by @Miguel Moreira do a good job of explaining the differences between the fields, but I just thought I'd provide some perspective that I can from having just finished a course on Algebraic Topology this past May. The big disclaimer here is that I have never taken a formal class in Differential Topology, but I can offer what I know about Algebraic Topology.
For my class in AT, we used the book written by William Massey. We pretty much just followed the book and ended the course with a study of these things called "covering spaces". But to answer your question, everything we did in that class involved some type of universal property and from talking to others, I'm not quite sure Differential Topology uses universal property notions as much.
We would define Free Abelian Groups as objects that satisfy this commutative diagram, and the hallmark theorem that is the Seifert-Van Kampen theorem was treated in this manner too. To be perfectly honest, what I remember most about that class is just seeing something like "oh, and this works because it satisfies this commutative diagram" absolutely everywhere, so really I feel jaded because all I associate with the subject of Algebraic Topology are commutative diagrams. I know it might sound silly to say but really everything in that class was very algebraic in nature (hence the name of the subject) and so we played around with fundamental groups a lot. While of course continuous functions were still in play (it is topology after all), we didn't care about differentiation on surfaces or manifolds, or if those manifolds were smooth or not. In fact, the word "smooth" never came up in the subject, and for good reason. Basically, for a first course on the subject, we just defined a few key concepts (manifolds, surfaces, homotopy, fundamental group, free groups, free abelian groups, SVK Theorem, covering spaces, local homeomorphisms) and studied how they could be useful when studying topological objects with an algebraic lens. Again, not sure how helpful this is, but this is just my two cents.