Separable manifolds (which I'll always assume to be Hausdorff) have partitions of unit (= are paracompact). This implies many nice properties.
E.g., if $V\to M$ is a vector bundle and $f,g:X\to M$ are two homotopic maps, then $f^*V\cong g^*V$.
All this fails for non-separable manifolds, as illustrated by the example of the long line $L$, whose tangent bundle is non-trivial but nevertheless there exists a (smooth) vector bundle over $L\times [0,1]$ whose restriction to $L\times \{0\}$ is the tangent bundle of $L$, and whose restriction to $L\times \{1\}$ is trivial.