For this answer, let us just work with $T_{0}$-spaces to avoid trivialities.

There exists regular spaces that are not completely regular, and in point-free topology, there exists regular frames that are not completely regular (frames are the point-free topological spaces). Furthermore, one can say that most regular spaces are not completely regular and that most regular frames are not completely regular depending on one's definition of mostness. However, I have not encountered any "naturally occurring" regular space or regular frame that is not completely regular.

The distinction between regularity and complete regularity is of philosophical importance. In general topology, there is a distinction between the "good spaces" that are used in analysis (such as manifolds, complete metric spaces, or even locally convex topological vector spaces) and the "bad spaces" (such as the cofinite topology, the Zariski topology, and non-Hausdorff spaces) (I put the word "bad" in quotes because I personally find these "bad" topological spaces to be quite interesting). One should therefore ask if there is an axiom that provides the dividing line between the "bad spaces" and the "good spaces".

In the past, I used to believe that complete regularity was the main separation axiom between the bad spaces and the good spaces, but now I think that there are just as good reasons to believe that regularity is the main separation axiom that distinguishes between the "bad spaces" and the "good spaces". In fact, regularity may be one cutoff between the "bad spaces" and the "good spaces" while complete regularity may be another cutoff, and there are only two distinct cutoffs because there are regular spaces that are not completely regular. Since regularity and complete regularity are different, the dividing line between "bad spaces" and "good spaces" may be a blur that spreads from regularity to complete regularity rather than a definite axiom.

Unlike Hausdorffness, regularity and complete regularity both extend seamlessly to point-free topology. Both regularity and complete regularity are very well-behaved in both general and point-free topology. They are both closed under taking products and subspaces.

The case for complete regularity as the cutoff.

A space is completely regular if and only if it can be embedded into a cube $[0,1]^{I}$ for some set $I$.

A space is completely regular if and only if it can be endowed with a compatible uniformity.

A space is completely regular if and only if it can be endowed with a compatible proximity.

The case for regularity as the cutoff.

A space $X$ is regular if and only if a filter $\mathcal{F}$ converges to a point $x_{0}$ precisely when the filter generated by $\{\overline{R}\mid R\in\mathcal{F}\}$ also converges to $x_{0}$.

There are several inequivalent ways of interpreting a topological space or frame in a forcing extension (or more generally, a larger model of ZFC). In any case, given a regular space $X$, if the forcing extension $V[G]$ collapses enough cardinals, then the interpretation of $X$ in $V[G]$ will be both regular and second countable.

A frame $L$ is regular if and only if there exists a frame $M$ such that the frame coproduct $L\oplus M$ is paracompact.

For this answer, let us just work with $T_{0}$-spaces to avoid trivialities.

There exists regular spaces that are not completely regular, and in point-free topology, there exists regular frames that are not completely regular (frames are the point-free topological spaces). Furthermore, one can say that most regular spaces are not completely regular and that most regular frames are not completely regular depending on one's definition of mostness. However, I have not encountered any "naturally occurring" regular space or regular frame that is not completely regular.

The distinction between regularity and complete regularity is of philosophical importance. In general topology, there is a distinction between the "good spaces" that are used in analysis (such as manifolds, complete metric spaces, or even locally convex topological vector spaces) and the "bad spaces" (such as the cofinite topology, the Zariski topology, and non-Hausdorff spaces) (I put the word "bad" in quotes because I personally find these "bad" topological spaces to be quite interesting). One should therefore ask if there is an axiom that provides the dividing line between the "bad spaces" and the "good spaces".

In the past, I used to believe that complete regularity was the main separation axiom between the bad spaces and the good spaces, but now I think that there are just as good reasons to believe that regularity is the main separation axiom that distinguishes between the "bad spaces" and the "good spaces". In fact, regularity may be one cutoff between the "bad spaces" and the "good spaces" while complete regularity may be another cutoff, and there are only two distinct cutoffs because there are regular spaces that are not completely regular. Since regularity and complete regularity are different, the dividing line between "bad spaces" and "good spaces" may be a blur that spreads from regularity to complete regularity rather than a definite axiom.

Examples of regular spaces which are not completely regular.

This paper by Mysior gives a quite simple example of a regular space that is not completely regular. This question also gives examples of regular spaces that are not completely regular.

Regularity and complete regularity are good axioms

Unlike Hausdorffness, regularity and complete regularity both extend seamlessly to point-free topology. Regularity behaves slightly better in this regard since the regularity axiom is a first order formula. Both regularity and complete regularity are very well-behaved in both general and point-free topology. They are both closed under taking products, subspaces, and sublocales in point-free topology.

The case for complete regularity as the cutoff.

A space is completely regular if and only if it can be embedded into a cube $[0,1]^{I}$ for some set $I$.

A space is completely regular if and only if it can be endowed with a compatible uniformity.

A space is completely regular if and only if it can be endowed with a compatible proximity.

The case for regularity as the cutoff.

A space $X$ is regular if and only if a filter $\mathcal{F}$ converges to a point $x_{0}$ precisely when the filter generated by $\{\overline{R}\mid R\in\mathcal{F}\}$ also converges to $x_{0}$.

There are several inequivalent ways of interpreting a topological space or frame in a forcing extension (or more generally, a larger model of ZFC). In any case, given a regular space $X$, if the forcing extension $V[G]$ collapses enough cardinals, then the interpretation of $X$ in $V[G]$ will be both regular and second countable.

A frame $L$ is regular if and only if there exists a frame $M$ such that the frame coproduct $L\oplus M$ is paracompact.

For this answer, let us just work with $T_{0}$-spaces to avoid trivialities.

There exists regular spaces that are not completely regular, and in point-free topology, there exists regular frames that are not completely regular (frames are the point-free topological spaces). Furthermore, one can say that most regular spaces are not completely regular and that most regular frames are not completely regular depending on one's definition of mostness. However, I have not encountered any "naturally occurring" regular space or regular frame that is not completely regular.

The distinction between regularity and complete regularity is of philosophical importance. In general topology, there is a distinction between the "good spaces" that are used in analysis (such as manifolds, complete metric spaces, or even locally convex topological vector spaces) and the "bad spaces" (such as the cofinite topology, the Zariski topology, and non-Hausdorff spaces) (I put the word "bad" in quotes because I personally find these "bad" topological spaces to be quite interesting). One should therefore ask if there is an axiom that provides the dividing line between the "bad spaces" and the "good spaces".

In the past, I used to believe that complete regularity was the main separation axiom between the bad spaces and the good spaces, but now I think that there are just as good reasons to believe that regularity is the main separation axiom that distinguishes between the "bad spaces" and the "good spaces". In fact, regularity may be one cutoff between the "bad spaces" and the "good spaces" while complete regularity may be another cutoff, and there are only two distinct cutoffs because there are regular spaces that are not completely regular. Since regularity and complete regularity are different, the dividing line between "bad spaces" and "good spaces" may be a blur that spreads from regularity to complete regularity rather than a definite axiom.

For this answer, let us just work with $T_{0}$-spaces to avoid trivialities.

There exists regular spaces that are not completely regular, and in point-free topology, there exists regular frames that are not completely regular (frames are the point-free topological spaces). Furthermore, one can say that most regular spaces are not completely regular and that most regular frames are not completely regular depending on one's definition of mostness. However, I have not encountered any "naturally occurring" regular space or regular frame that is not completely regular.

The distinction between regularity and complete regularity is of philosophical importance. In general topology, there is a distinction between the "good spaces" that are used in analysis (such as manifolds, complete metric spaces, or even locally convex topological vector spaces) and the "bad spaces" (such as the cofinite topology, the Zariski topology, and non-Hausdorff spaces) (I put the word "bad" in quotes because I personally find these "bad" topological spaces to be quite interesting). One should therefore ask if there is an axiom that provides the dividing line between the "bad spaces" and the "good spaces".

In the past, I used to believe that complete regularity was the main separation axiom between the bad spaces and the good spaces, but now I think that there are just as good reasons to believe that regularity is the main separation axiom that distinguishes between the "bad spaces" and the "good spaces". In fact, regularity may be one cutoff between the "bad spaces" and the "good spaces" while complete regularity may be another cutoff, and there are only two distinct cutoffs because there are regular spaces that are not completely regular. Since regularity and complete regularity are different, the dividing line between "bad spaces" and "good spaces" may be a blur that spreads from regularity to complete regularity rather than a definite axiom.

Unlike Hausdorffness, regularity and complete regularity both extend seamlessly to point-free topology. Both regularity and complete regularity are very well-behaved in both general and point-free topology. They are both closed under taking products and subspaces.

The case for complete regularity as the cutoff.

A space is completely regular if and only if it can be embedded into a cube $[0,1]^{I}$ for some set $I$.

A space is completely regular if and only if it can be endowed with a compatible uniformity.

A space is completely regular if and only if it can be endowed with a compatible proximity.

The case for regularity as the cutoff.

A space $X$ is regular if and only if a filter $\mathcal{F}$ converges to a point $x_{0}$ precisely when the filter generated by $\{\overline{R}\mid R\in\mathcal{F}\}$ also converges to $x_{0}$.

There are several inequivalent ways of interpreting a topological space or frame in a forcing extension (or more generally, a larger model of ZFC). In any case, given a regular space $X$, if the forcing extension $V[G]$ collapses enough cardinals, then the interpretation of $X$ in $V[G]$ will be both regular and second countable.

A frame $L$ is regular if and only if there exists a frame $M$ such that the frame coproduct $L\oplus M$ is paracompact.