Why do topologists dislike sequences? Okay, the header might be a slight exaggeration, but...
My introductory text on point-set topology defines convergence of point sequences in topological spaces, but then immediately states that these do in fact not play a large role in the theory of topological spaces, contrary to $\Bbb R^n$. In fact, it claims that much of the motivation behind the development of point-set topology was to get rid of the great reliance on point sequences.
But why this motivation? Why would you want to get rid of a tool as invaluable as point sequences?
 A: Because the usual (and useful properties) of sequences in metric spaces are false in topological spaces without more hypothesis. Only one example: sequential continuity isn't equivalent to continuity.
A: It is not that one feels the need to "get rid of" sequences. You are certainly right that sequences are useful, for the simple reason that many, many topological spaces are metric spaces, and sequences can be used to detect closure of subsets of a metric space. Sequences remain useful even to topologists at the current boundary of research.
Nonetheless, there do exist topological spaces that are not metric, and for which sequences are not sufficient to detect closure. These topological spaces arise naturally, and it is necessary to understand them for many applications. So one needs to develop a theory to understand them. Hence the development of separation axioms and associated topics of point set topology. Then, of course, those topics become intellectually interesting in and of themselves, hence the further growth of point set topology.
Here is an example, an extremely non-Hausdorff quotient space of the torus $S^1 \times S^1$. Consider the flow $t \cdot (z,w) = (e^t z, e^{\sqrt{2}t}w)$, defined for $t \in \mathbb{R}$. Two points $(z,w)$, $(z',w')$ are on the same "flow line" if there exists $t$ such that $t \cdot (z,w) = (z',w')$. This is an equivalence relation, and so flow lines form a decomposition of the torus. Each flow line is the injective image of a line of slope $\sqrt{2}$ in the universal cover $\mathbb{R} \times \mathbb{R}$. Every flow line is dense in $S^1 \times S^1$. Therefore in the quotient space of $S^1 \times S^1$---the one that is defined using the decomposition into flow lines---every point is dense. Sequences are quite useless for exploring the topology of the quotient.
A: Even in everyday calculus, or in ${\mathbb R}^n$, sequences should only be used for the construction of new and interesting objects, like $\exp$ or $\sqrt{123}$, but not as a didactic tool for understanding limits or continuity. These notions are difficult enough to grasp for many in their original version. Therefore it doesn't make sense to load them up with at least three more nested $\forall$'s and $\exists$'s, as is the case when limits and continuity are explained in terms of sequences.
A: Somewhere in Munkres' topology textbook there is (or was in an earlier edition?) a set of exercises that prove that nets do for topological spaces generally what sequences do for metric spaces and other spaces satisfying the first axiom of countability.
Here are three familiar examples of convergence of a net:


*

*$\lim\limits_{n\to\infty}\dfrac 1 n = 0$.

*$\lim\limits_{x\to0}\dfrac{\sin x}{x}=1$.

*$\displaystyle\lim_{\|P\|\to0} \sum_P f(x)\,\Delta x=\int_0^1 f(x)\,dx$ where $\|P\|\to0$ means the mesh of the partition $P$ approaches $0$ and the sum is over all intervals in the partition, and $x$ is in the interval in question, and $\Delta x$ is the length of that interval.


The third example above is the only instance of convergence of a net that is familiar to everyone who's never taken a topology course and is not either a limit of a sequence or a limit of a function of the kind treated in the usual freshman calculus course.
PS: In the exercises in Munkres, it is useful to use as the directed set whose members are the indices in the net, the set of all open neighborhoods of the point to which the net is to converge, ordered by the inverse of inclusion.
