I'm a third year undergraduate student and I had a basic course on Algebraic Topology which covered chapters 1-6 of Armstrong's Basic Topology, which consists of Homotopy, Fundamental Group, Simplicial Complexes and van Kampen theorem. I'm going to do a project on Teichmüller Theory and I need to learn some Homology and Cohomology. Also I want to gain a deeper understanding of Algebraic Topology in general.
In my way to find a good book to study Algebraic Topology, I didn't find Hatcher's book interesting because of its style of exposition but since I've had graduate courses in Complex Analysis and Smooth Manifolds, Fulton's Algebraic Topology seemed really interesting to me because of its approach which uses Analysis and Geometry as motivation for developing Algebraic Topology. But on the other hand, I've heard that Fulton's approach is a bit old and some other books such as Rotman's An Introduction to Algebraic Topology pursue a more modern and categorical viewpoint (Btw Rotman's book seemed good to me).
Now, I'm a bit stuck between Fulton's book and Rotman's book. Is there any significant difference between these books' approaches? Should I study Rotman's or Fulton's? Is there any better book than these two?