Are neighbourhood and open neighbourhood always exchangeable in statements? From Wikipedia

If $X$ is a topological space and $p$ is a point in $X$, a
  neighbourhood of $p$ is a subset $V$ of $X$, which includes an open
  set $U$ containing $p$,
Note that the neighbourhood $V$ need not be an open set itself. If $V$
  is open it is called an open neighbourhood. Some authors require that
  neighbourhoods be open, so it is important to note conventions.

Clearly  neighbourhood and open neighbourhood are two different concepts. But in the limited number of statements I have seen and can recall, neighbourhood and open neighbourhood can always replace each other without changing the statements from true to false, or from false to true.
So I was wondering if neighbourhood and open neighbourhood can always be exchangeable in statements? If not always, is it most of the cases? What are some statements where exchanging between neighbourhood and open neighbourhood matters? Well, a trivial example is: "an open neighbourhood is a neighbourhood" is true while "a neighbourhood is an open  neighbourhood" isn't. But these exemplar statements  are not really meaningful.
What is the purpose of distinguishing between neighbourhood and open neighbourhood?
Thanks and regards!
 A: There is usually no point in distinguishing -- because mathematicians tend not to use the word "neighborhood" in situations where it would make a difference whether it is supposed to be open.
More concretely, whenever one uses the statement "there exists a neighborhood $A$ of $x$ such that $A$ has the property $P$", then it is almost always the case that if such a non-open neighborhood existed, there would also be an open subneighborhood of it with property $P$ anyway. When $P$ is such that this doesn't hold, "neighborhood" would feel like a wrong term to use.
A: Very often you don't care about using one or the other, as you say. But of course there is a difference and of course some statements are sensitive to this difference.
For example, local compactness is an interesting property of topological spaces. And in most of the cases we deal with, an open neighborhood will not be compact, so we need the more general definition of neighborhood. So a concrete example is:

Every point in $\mathbb{R}$ has a compact neighborhood.

You see here that you cannot say compact open neighborhood, because then the statement becomes false.
Another interesting property of topological spaces is regularity ($T_{3}$ separation axiom). And in regular spaces, we have:

If $X$ is a regular topological space and $x\in X$ is a point, then any neighborhood $U$ of $x$ contains a closed neighborhood $C$ of $x$.

Again, you cannot say closed open neighborhood (if our space is connected, say). So you see that there are indeed some examples, and not very pathological or weird, in which it is important to make a distiction between neighborhood and open neighborhood.
