I am reading the book Introduction to Smooth Manifolds by John Lee. In his book he proves a theorem called the Whitney's Approximation theorem which essentially states that any continuous map can be approximated by a smooth map.
In the end he gives an application where he proves that any homotopy between smooth manifolds is an isotopy. I am wondering if it has more applications.Thank You.
What you mean in the last paragraph that two homotopic smooth maps are smoothly homotopic. I find this a pretty pleasant fact, that "smooth homotopy theory" on manifolds is the same thing as continuous.
Smooth maps are well-behaved, much better than continuous maps. Here's a proof of the Brouwer fixed point theorem without any algebraic topology.
Suppose I have a map $f: D^n \to S^{n-1}$ that's the identity on the boundary. Then, using a version of Whitney's theorem, I may as well assume this map is smooth. Pick a regular value $x \in S^{n-1}$; these exist by Sard's theorem. Then by a version of the regular value theorem $f^{-1}(x)$ is a properly embedded (that is, $\partial f^{-1}(x) = f^{-1}(x) \cap \partial D^n$) compact submanifold of dimension 1. Thus it has an even number of endpoints. But the very fact that $f$ is the identity on the boundary means that $f^{-1}(x) \cap \partial D^n = \{x\}$ - which is one point. Contradiction. Hence Brouwer's.
You can imagine that other arguments can also benefit from the assumption that the map is smooth.