This is a question about first countable spaces. Topology of such spaces can be defined in terms of convergent sequences, and many topological properties of such spaces can be expressed in terms of sequences by taking general theorems about nets and replacing the term "net" with the term "sequence". For example: A set X is closed iff the limit of any convergent sequence in X belongs to X.
However, this is not true for compactness. If we take a general description of compactness in terms of nets and replace "net" with "sequence", we obtain a statement that is true only in second countable (or in Lindelöf) spaces: X is compact iff every sequence in X has convergent subsequence, which limit belongs to X.
So, the first question is, why compactness is different in this respect?
And the second question: is it possible to somehow adapt the above statement to make it true for all first countable spaces?
EDIT: Steven's answer is very good, but it answers a slightly different question. So I'll try to rephrase my question more precisely.
(1) We know that in general topological spaces the openness and closedness of sets can be expressed in terms of convergent nets.
(2) Therefore, any topological property can be expressed in terms of converged nets.
(3) We know that in the first countable spaces the openness and closedness of sets can be described in terms of convergent sequences.
(4) Therefore, in the first countable spaces, any topological property can be described in terms of converged sequences.
But, how to obtain such a description? The obvious thing that comes to mind, and which indeed in many cases produces a true statement in the first countable spaces, is
(5) Take a known characteristic of a topological property in terms of nets and replace each occurrence of "net" with "sequence".
However, there are cases where action (5) doesn't produce a true statement. For example,
(6) Application of action (5) to compactness results in a statement which, to be true in the first countable spaces, requires additional restrictions (e.g., Lindelöf condition).
On the other hand, in pseudometric spaces action (5) works well for both compactness and completeness, and seems to be universally applicable. So the question:
Is there a property, that is absent in the first countable spaces and therefore prevents the universal applicability of action (5), but is present in pseudometric spaces, where it ensures such universal applicability?
First, we observe that among first countable spaces, every compact is sequentially compact: https://topology.pi-base.org/spaces/?q=First+Countable%2BCompact%2B%7ESequentially+Compact
But the converse fails: https://topology.pi-base.org/spaces/?q=First+Countable%2B%7ECompact%2BSequentially+Compact The particular counterexample at hand is the first uncountable ordinal: each point is first countable, and every sequence is bounded and thus converges. However, an open cover of sets of countable size has no finite sub cover.
As you've observed, strengthening first countable (countable local weight) to second Countable (countable global weight) is sufficient to guarantee compactness from sequential compactness: https://topology.pi-base.org/spaces?q=Second+Countable%2B%7ECompact%2BSequentially+Compact What's really happening here is that you have both a property (second countable) sufficient to guarantee all covers have countable subcovers, and a property (sequentially compact) sufficient for all countable covers to have finite subcovers.
Here's an open (to pi-Base) question: does there exist a first countable (or not), weakly Lindelöf, sequential compact space that fails to be compact? https://topology.pi-base.org/spaces?q=weakly+lindelof%2BSequentially+Compact%2B%7ECompact
Well https://topology.pi-base.org/spaces/S000040 is the desired counter example; it's just missing its weakly Lindelöf property. It can be modified to be first countable; I'll write up details at a computer later, but I think it can be done to make things $T_1$, but perhaps not $T_2$.