Topology vs Borel sigma-algebra on a set $X$

They're not the same. If you have a topology (only the open sets) you get a Borel $\sigma$-algebra: the smallest one containing all open sets (so that contains the topology).

But this also contains all closed sets, all countable intersections of open sets (which need not be open, nor closed) etc. The Borel $\sigma$-algebra will generally be a lot bigger.

Also note that $\sigma$-algebras are closed under complements (!), countable unions and countable intersections. While topologies are closed under finite intersections and arbitrary unions. So quite different.

Any space with a topology automatically has a Borel $\sigma$-algebra (if we need it, say for measure theory), while the having a $\sigma$-algebra does not mean having a topology.

A $\sigma$ algebra needs to be closed under ${\bf countable}$ union and under complementation. Now, a topology is closed under ${\bf any}$ union and in general not closed under complementation.

This holds in general. Furthermore, you have that the Borel $\sigma$ algebra contains the topology (by definition), hence it contains the closed sets too, so that in general those two objects are different.