Difference between subbasis and cover in topology In the wikipedia page for covers it is stated that:

Covers are commonly used in the context of topology. If the set $X$ is
a topological space, then a cover $C$ of $X$ is a collection of subsets $U_\alpha$
of $X$ whose union is the whole space $X$.

On the other hand, Munkres second edition, page 82, chapter 2, says that:

A subbasis $\mathcal{S}$ for a topology on $X$ is a collection of
subsets of $X$ whose union equals $X$. The topology generated by the
subbasis $\mathcal{S}$ is defined to be the collection $\mathcal{T}$
of all unions of finite intersections of elements of $\mathcal{S}$

I understand from these two definitions that a subbase/subbasis is the same as a cover, with the caveat that maybe the name "subbasis" is used in a context in which we will want to generate a topology from it. Is this correct or is there any other subtle difference I'm missing?
 A: Your are basically correct, except that the restricted context of sub-basis is quite a bit more restrictive than you seem to be implying. Also, you are missing an important followup definition, which is the concept of a sub-basis for a given topological space.
To start with, in the context of a bare set $X$ with no topology specificied, a sub-basis-for-a-topology-on $X$ is indeed just a lot of words for a cover of the set $X$. Notice: no actual topology is involved... yet... Now one goes on to specify exactly which topology is generated by a given sub-basis-for-a-topology-on-$X$, namely the collection all unions of finite intersections of elements of that sub-basis.
Consider now a different context, namely a topological space, which means a set $X$ with a specified topology $\mathcal T$ that is already given. In this context one wants to be able to talk about a sub-basis for the topology $\mathcal T$. For that purpose one needs the following very simple fact:

A cover $\mathcal S$ of $X$ generates the topology $\mathcal T$ of $X$ if and only if every element of $\mathcal S$ is an element of $\mathcal T$, and every element of $\mathcal T$ can be written as a union of finite intersections of elements of $\mathcal S$.

And then one follows that up with the important definition:

A sub-basis for a topological space $X$ is a cover of $X$ that generates the topology on $X$.

Notice that the topology is not named in that last definition, because it is already implicit: $X$ is given to us as a topological space, i.e. a set with a topology.

One could summarize the situation by saying that the concept of sub-basis-for-a-topology is used in two different (but related) ways: to construct a new topology on a set $X$ by choosing a cover and using the topology generated by that cover; and to study a given topology on a set $X$ by using a cover which generates that given topology.
