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.