Difference between Measurable and Borel Measurable function

A Borel measurable function is a measurable function but with the specification that the measurable space $X$ is a Borel measurable space (where $\mathfrak B$ is generated as the smallest sigma algebra that contains all open sets). The condition "$f$ is continuous" is equivalent to "$f^{-1}(V)$ is open (and thus Borel measurable) for every open set $V\subseteq Y$".

But not every measurable function is Borel measurable, for example no function that takes arguments from $(\mathbb R,\{\emptyset,\mathbb R\})$ is Borel measurable, because $\{\emptyset,\mathbb R\}$ is not a Borel sigma algebra.

The difference is in the $\sigma$-algebra that is part of the definition of measurable space. In your first definition you say that $X$ is measurable but you don't specify the $\sigma$-algebra; in the second you specify that the $\sigma$-algebra is the collection of all Borel sets.

The $\sigma$-algebra is a collection of sets closed under countably-infinite unions (that's rather rough and ready, but if you need to know more you're better off looking up a definition and some examples).