Suppose I have two vector bundles E \rightarrow M, E' \rightarrow M of rank $k$ on a smooth manifold $M$. Let \mathcal{E}(M), \mathcal{E'}(M) denote their spaces of smooth sections. We can choose some arbitrary isomorphism \phi_p: E_p \rightarrow E'_p for all $p \in M$, where E_p, E'_p denote the fibers above $p$.
Now we use this to define a map \mathcal{F}: \mathcal{E}(M) \rightarrow \mathcal{E'}(M) as follows. For any smooth section $\sigma \in \mathcal{E}(M)$, define the section $\mathcal{F}(\sigma)$ by $\mathcal{F}(\sigma)(p) = \phi_p(\sigma(p))$. Then $\mathcal{F}$ is linear over $C^\infty(M)$, so there is a smooth bundle map F: E \rightarrow E' over $M$ such that $\mathcal{F}(\sigma) = F \circ \sigma$ for all $\sigma$. Defining a map \mathcal{F}^{-1}: \mathcal{E}'(M) \rightarrow \mathcal{E}(M) using $\phi_p^{-1}$, we see by the same reasoning that there is a smooth bundle map F^{-1}: E' \rightarrow E which is the inverse of $F$. So the two bundles are isomorphic.