The statement: differentiable complex line bundles are determined by their Chern class in $H^2(X,\mathbb{Z}$) is from Wells' book p.105, which I attached below (also the statement of theorem 4.5 is attached).
I feel very confused about this statement. My understanding is fist we use that there is a one-to-one correspondence between the equivalence classes of differentiable line bundles on $X$ and the elements of the cohomology group $H_1(X,\epsilon^*)$, where $\epsilon^*$ means non-vanishing differentiable structure sheaf. And by the isomorphism of $H^1(X,\epsilon^*)\cong H^2(X,\mathbb{Z})$, we can determine the equivalent classes of equivalence classes of differentiable line bundles via the elements of $H^2(X,\mathbb{Z})$. However, the Chern classes are in $H^2(X,\mathbb{R})$ or more precisely, $\tilde{H}^2(X,\mathbb{Z})$, where $\tilde{H}^2(X,\mathbb{Z})$ is the image of $H^2(X,\mathbb{Z})$ from the homomorphism from $H^2(X,\mathbb{Z})\to H^2(X,\mathbb{R})$ induced by the inclusion of $\mathbb{Z}\to\mathbb{R}$ of constant sheaf. But we don't have that Chern classes are in $H^2(X,\mathbb{Z})$. I don't understand how to determine the line bundles via the classes are in $H^2(X,\mathbb{Z})$.