The title pretty much says it all. As far as I can tell, the terms "inclusion" and "monomorphism" are equivalent. (Ditto for $\hookrightarrow$ and $\rightarrowtail$.) Is this the case?
Edit: Judging by the answer and comments I've received so far, I realize I'm working with definitions that may not be universal. My source is Mitchell's Theory of Categories (1965). On p. 6, he writes (emphasis in the original):
If $\alpha:A' \to A$ is a monomorphism, we shall call $A'$ a subobject of $A$, and we shall refer to $\alpha$ as the inclusion of $A'$ in $A$.
As far as this quote goes, I'd say that "inclusion" and "monomorphism" are essentially synonymous, differing only in the fine grammatical structures of the English phrases that can be formed with each; e.g. "$f$ is the inclusion of $X$ in $Y$" vs. "$f$ is a monomorphism from X to Y". The fact that Mitchell uses the definite article for "the inclusion", even though it was defined by the existence of "a monomorphism" (indefinite article), is a bit disconcerting, but Mitchell addresses this concern a few lines later:
...it is important to remember that in general there is more than one monomorphism from $A'$ to $A$, and that whenever we speak of $A'$ as a subobject of $A$ we shall be referring to a specific monomorphism $\alpha$.
On the basis of this additional clarification, I'd "fill out" Mitchell's definition of "inclusion" slightly: if $A'$ is a subobject of $A$ (as defined above), then the inclusion of $A'$ in $A$ is
the monomorphism $\alpha$ that our designation of $A'$ as a subobject of $A$ is based on. (✽)
In fact, I imagine that, given this context, one could use a phrase like "the monomorphism of $A'$ to $A$" as shorthand for the formula (✽) I just proposed. If so, "inclusion" and "monomorphism" would be quite synonymous. (At the very least, there is a bijection between the class of monomorphisms and the class of inclusions.)
Due to these considerations, I was puzzled by random comments I'd found online that implied some meaningful distinction between these two terms (e.g. statement of preference as to which type of arrows to assign in diagrams to inclusions and which to monomorphisms), and this prompted my original question.
I am now particularly interested in learning of examples of monomorphisms that are not inclusions, as alluded to by Zhen Lin in one of the comments below. This will give me a clue about the various non-synonymous meanings given to the term "inclusion".
I regret that this clarification of my original post is many times longer than the original, but after having erred perhaps on the side of brevity, now I'm making up for it by going verbose.