is a net stronger than a transfinite sequence for characterizing topology?
The spaces characterized by the property that a subset is closed if and only if it is closed w.r.t. limits of transfinite sequences are called pseudoradial spaces.
I would like to give an example of a space that is not pseudoradial together with sketch of a proof. (I hope I am not doing something very easy in a too complicated way.) I would also like to mention a few references and some properties of these spaces.
Before presenting the example, one short remark. Pseudoradial spaces are represented by the convergence of net on well-ordered nets. Obviously, it is sufficient to take the nets on ordinals. We can go a little further - regular cardinals are sufficient. Indeed, if we have a cofinal subset of an ordinal, we can use this cofinal subset to get another convergent net.
Let us consider the following example. Each arrow in the picture bellow represents a convergent sequence. I.e., this is a topological space homeomorphic to $\{0\}\cup\{\frac1n;n\in\mathbb N\}$ taken as a subspace of real line. Equivalently, this is precisely the ordinal $\omega+1$ taken with the order topology.
We take all these sequences and identify some of the points as in the picture. (I.e., we make a quotient space of some of these spaces.) Let us call the resulting space $S_2$. Then we take the subspace of this space as shown in the picture. This subspace will be called $S_2^-$. (I've taken the notation $S_2$ and $S_2^-$ from this paper: Franklin S.P., Rajagopalan M., On subsequential spaces, Topology Appl. 35 (1990), 1-19. But you can notice that this space is very very similar to Arens-Fort space mentioned in Brian's answer.)
Now we want to show that $S_2^-$ is not pseudoradial.
Note that the space $S_2^-$ has only one non-isolated point. Let us call it $\omega$ . So we ask whether there is a transfinite sequence, consisting only of points different from $\omega$, which converges to $\omega$.
First, let us show that this is not possible for a regular cardinal $\alpha>\omega$. Suppose that $(x_\eta)_{\eta<\alpha}$ is an $\alpha$-sequence of points of $S_2^-\setminus\{\infty\}$, which converges to $\infty$. Let us denote $n_\eta$ the "column" to which $x_\eta$ belongs. In we use the notation the notation from the picture bellow $n_\eta$ is the first coordinate of ordered pair $x_\eta$.
We can see that $n_\eta$ converges to $\omega$. (E.g. by noticing that $(x,y)\mapsto x$ and $\omega\to\omega$ is a quotient map from $S_2^-$ to $\omega$ with order topology.)
Now this is not possible, since the we would be able to construct an increasing $\alpha$-sequence converging to $\omega$ and using this sequence we would be able to show that cofinality of $\alpha$ is $\omega$.
So the only possibility is to take a sequence in the usual sense, i.e., a sequence of length $\omega$. Perhaps with a little handwaving, but it is more-or-less clear that general situation is similar to the situation when the $n$-term of the sequence is in the $n$-th column. So we have a sequence $x_n=(n,y_n)$. Obviously $\{\omega\}\cup\bigcup\limits_{n\in\omega} \{n\}\times(y_n,\infty)$ is a neighborhood of the point $\omega$ containing no terms of this sequence.
Pseudoradial spaces were introduced by H. Herrlich. Quotienten geordneten Räume und Folgenkonvergenz. Fund. Math., 61:79–81, 1967; pdf. They were later studied by A.V. Arhangelskii and many others.
The class of pseudoradial spaces is closed under the formation of closed subspaces, quotients and topological sums. They are a coreflective subcategory of the category Top of all topological spaces. This means that for each topological space we have pseudoradial coreflection; a pseudoradial space which is, in some sense, close to this space. The pseudoradial coreflection is obtained simply by taking sets closed under limits of transfinite sequences as closed sets in a new topology on the same set. (E.g. the pseudoradial coreflection of $S_2^-$ is discrete.)
The same thing can be done with any class $\mathbb P$ of directed sets instead of ordinals. This is called $\mathbb P$-net spaces in P. J. Nyikos. Convergence in topology. (In M. Hušek and J. van Mill, editors, Recent Progress in General Topology, pages 537–570, Amsterdam 1992. North-Holland.) The properties of pseudoradial spaces which I mentioned in the preceding paragraph are true for $\mathbb P$-net spaces, too.
Interestingly, if we take the linearly ordered sets, we obtain the same class of spaces as from well-ordered sets, see James R. Boone: A note on linearly ordered net spaces. Pacific J. Math. Volume 98, Number 1 (1982), 25-35; link.
The answer to your last question is yes. $\beta\omega$, the Čech-Stone compactification of $\omega$, is such a space. No sequence in $\omega$, transfinite or otherwise, converges unless it’s eventually constant, but each point in $\beta\omega\setminus\omega$ is the limit of a net in $\omega$. Specifically, if $p\in\beta\omega\setminus\omega$, we can regard $p$ as a free ultrafilter on $\omega$. Let $I=\{\langle n,A\rangle:n\in A\in p\}$, and let $\nu:I\to\omega:\langle n,A\rangle\mapsto n$; then $\nu$ is a net that converges to $p$.
A more elementary example is the Arens-Fort space, which is a countable Hausdorff space that is not first countable at its one non-isolated point. Let $X$ be this space, let $p$ be the one non-isolated point, and let $D$ be the set of isolated points. Then no sequence in $D$ of any length converges to $p$, but the net in $D$ built in the usual way from the nbhds of $p$ does converge to $p$.