Under what circumstances does Heine continuity differ from ordinary continuity In what kinds of topological spaces are the functions that preserve limits of sequences different from the continuous functions?

Let $X$ and $Y$ be topological spaces. I will restrict attention to total functions only, so $f : X \to Y$ is defined everywhere.
A function $f : X \to Y$ is defined to be continuous if, for every open set $V$ in $Y$, $f^{\leftarrow}(V)$ is open in $X$.
I was thinking earlier today about limits of sequences and how to define limits in topological terms. I think the following works:
$$ \lim_{n \to \infty} \vec{a}_n = z \;\; \text{iff}\;\; \text{for every open set $U \ni z$, there exists a $k$ such that $\{a_k, a_{k+1}, \cdots\}$ is a subset of $U$} $$
Anyway, I was wondering if preserving limits of sequences is equivalent to continuity.
$$ \lim_{n \to \infty} f(a_n) = f\left(\lim_{n \to \infty} a_n\right) $$
I tried looking it up and came across the following interesting section on Wikipedia. This section covers settings more general than $\mathbb{R}$ and contains the following interesting claim.

In several contexts, the topology of a space is conveniently specified in terms
of limit points. In many instances, this is accomplished by specifying when a
point is the limit of a sequence, but for some spaces that are too large in some
sense, one specifies also when a point is the limit of more general sets of
points indexed by a directed set, known as nets. A function is
(Heine-)continuous only if it takes limits of sequences to limits of sequences.
In the former case, preservation of limits is also sufficient; in the latter, a
function may preserve all limits of sequences yet still fail to be continuous,
and preservation of nets is a necessary and sufficient condition.

I'm struggling to visualize an example where the limits of sequences isn't informative enough to characterize continuity.
I'm only familiar with the notion of a net only from the punchline "it's a generalization of a sequence" and "its domain is a directed set". I haven't seen a context yet that motivates the notion of a net (besides this one).
Is there a simple way to characterize the spaces where Heine-continuity and continuity are not equivalent?
 A: This is dependent on whether the topology on $X$ is so-called sequential, i.e. can be described completely using convergent sequences.
A set $A \subseteq X$ is called sequentially closed iff for all sequences $(a_n)_n$ in $A$ (i.e. $a_n \in A$ for all $n \in \Bbb N$) such that $a_n \to x$ in $X$ we can conclude that $x \in A$ as well.
In every topological space $X$ a closed subset is sequentially closed but if the converse holds (every sequentially closed $A$ is closed in $X$) then $X$ is called a sequential space. So we only need to know what sequences converge in $X$ to know what all closed subsets (and hence all open subsets too) in $X$ are. Most common spaces are sequential, i.e. all first countable (with a countable local base at every point) spaces are sequential, so all metric spaces are. This "explains" why sequence arguments are so common in analysis (where we mostly work with metric spaces) and sufffice to show continuity etc.

Theorem: if $f: X \to Y$ is sequentially continuous (what you call Heine-continuous) and $X$ is a sequential space, then $f$ is continuous.

A simple to describe space that is not sequential is the co-countable topology on $\Bbb R$ (a set $A$ is closed iff $A=\Bbb R$ or $A$ is at most countable), and another is $\Bbb R^I$ where $I$ is of cardinality continuum. Some more (Cech-Stone compactifications e.g.) exist. This motivates the introduction of nets where we can show in any space that closed="closed under net-limits" and continuous = "net-continuous", etc.
So $X$ being sequential is key for your continuity.
