I highly recommend Chapter 16 of Eric Schechter's Handbook of Analysis and its Foundations (AP, 1997), which deals with separation axioms and their individual importance in quite a bit of detail. One reason that separation axioms are implicit (rather than implicitexplicit) in many mathematical discussions outside general topology is that basic topological spaces (such are $\mathbb{R}^n$) that we use to construct other spaces already satisfy all most of the separation axioms, and these properties survive these constructions so do not need to be checked separately (key words here are hereditary, productive, initial).
If you squint at some of the separation axioms, you'll notice they mostly have to do with pairs of points and pairs of open/closed sets containing them. Thus, separation properties of a topological space $X$ can be naturally expressed as topological properties of the Cartesian product space $X\times X$ and its diagonal subset, $X\to X\times X$. Recall that relations on $X$ are subsets of $X\times X$, so one might expect some non-trivial interplay between relations on topological spaces (like orders or equivalence relations) and separation axioms. For instance, some separation properties fail to survive the quotient of a topological space by an equivalence relation. Then they have to be discussed explicitly. For example, famously the space of leaves of a smooth foliation of a manifold may fail to be Hausdorff.
I highly recommend Chapter 16 of Eric Schechter's Handbook of Analysis and its Foundations (AP, 1997), which deals with separation axioms and their individual importance in quite a bit of detail. One reason that separation axioms are implicit (rather than implicit) in many mathematical discussions outside general topology is that basic topological spaces (such are $\mathbb{R}^n$) that we use to construct other spaces already satisfy all most of the separation axioms, and these properties survive these constructions so do not need to be checked separately (key words here are hereditary, productive, initial).
If you squint at some of the separation axioms, you'll notice they mostly have to do with pairs of points and pairs of open/closed sets containing them. Thus, separation properties of a topological space $X$ can be naturally expressed as topological properties of the Cartesian product space $X\times X$ and its diagonal subset, $X\to X\times X$. Recall that relations on $X$ are subsets of $X\times X$, so one might expect some non-trivial interplay between relations on topological spaces (like orders or equivalence relations) and separation axioms. For instance, some separation properties fail to survive the quotient of a topological space by an equivalence relation. Then they have to be discussed explicitly. For example, famously the space of leaves of a smooth foliation of a manifold may fail to be Hausdorff.