Smooth homotopy theory

Yes, the map you mention is an isomorphism. I think the main reason people rarely address your specific question in literature is that the technique of the proof is more important than the theorem. All the main tools are written up in ready-to-use form in Hirsch's Differential Topology textbook.

There are two steps to prove the theorem. Step 1 is that all continuous maps can be approximated by (necessarily) homotopic smooth maps. The 2nd step is that if you have a continuous map that's smooth on a closed subspace (say, a submanifold) then you can approximate it by a (necessarily) homotopic smooth map which agrees with the initial map on the closed subspace. So that gives you a well-defined inverse to your map $\phi$.

There are two closely-related proofs of this. Both use embeddings and tubular neighbourhoods to turn this into a problem for continuous maps defined on open subsets of euclidean space. And there you either use partitions of unity or a "smoothing operator", which is almost the same idea -- convolution with a bump function.

Dear Paul, as Ryan says the smooth and continuous homotopy groups of a manifold coincide.

This is stated as Corollary 17.8.1 in Bott and Tu's book Differential Forms in Algebraic Topology (Springer Graduate Texts in Mathematics, #82).The Corollary is to the preceding Proposition 17.8, which says that a continuous map is homotopic to a differentiable one.This is easy but relies on Whitney's embedding theorem for which the authors refer to De Rham's book Variétés différentiables ; you might prefer Hirsch's book in the same Springer series, GTM #33, which is more modern and in English.

As an application Bott and Tu calculate $\pi_q S^n$ for $q\leq n$ by differential methods.