The standard reference is probably still the book Vector Measures by Diestel and Uhl.
If $f$ is the pointwise limit of Borel measurable functions, then $f$ is Borel measurable. To see this, note that it suffices to show that the preimage of an open set is measurable and if $\langle f_n\rangle$ is a sequence of Borel measurable functions converging pointwise to $f$ and $O$ is open, then
$$f^{-1}(O)=\{\omega:f(\omega)\in O\}=\{\omega:\lim_n{ f_n(\omega)\in O}\}$$
$$=\{\omega:f_n(\omega)\in O\text{ for }n\text{ large enough}\}=\bigcup_{n=1}^\infty\bigcap_{m=n}^\infty f_m^{-1}(O).$$
In particular, Bochner measurable functions are measurable. In a separable Banach space, every Borel measurable function will be the pointwise limit of simple functions. Just pick a countable dense set $D=\{d_1,d_2,\ldots\}$ and for $f$ Borel measurable let $f_n$ have value $d_m$ on $f^{-1}(B_{1/n}(d_m))$ for $m\leq n$ and value $0$ everywhere else. Then $\langle f_n\rangle$ converges pointwise to $f$. If one is willing to allow for simple functions with countably many values, as is sometimes done, one can make the convergence uniform.
If $f$ is the pointwise limit of simple functions, then the range of $f$ is in the closure of the set of values of the simple functions, so a Bochner measurable function must have values in a separable subspace. If $X$ is a nonseparable Banach space then the identity from $X$ endowed with the Borel $\sigma$-algebra to $X$ is Borel measurable, but not Bochner measurable.
Many of these concepts also come with almost-everywhere provisos instead of the pointwise version, but the difference is inessential if one works with complete measure spaces.
