"And if you are in doubt about what We have revealed down upon Our servant, then produce a surah like thereof and call upon your witnesses other than Allah- if you should be truthful. But if you do not- and you will never be able to- then fear the Fire, whose fuel is men and stones, prepared for the disbelievers." [Q 2:23-24]
This and similar verses represent how the Qur'an introduces its miracle: in the form of a challenge to humanity which would remain forever uncontested. The claim of inimitability.
A miracle, simply and simplistically (for our purposes) defined, would mean an event which cannot be accounted for by referring to the natural causes relevant to that time and place. The parting of the Red Sea, for example, couldn't have been accounted for by the conjunction of all the relevant natural causes e.g. waves, air current, force of the staff etc, and as such would count as a miracle.
A claim of inimitabilty, on the other hand, seems stronger than a miracle claim, at least in the following way. The fact that thing X is inimitable presents itself to someone's mind only after she realizes that thing X is a miracle. In other words, the chronology of her thoughts would look something like this:
Consideration 1. Thing X couldn't have been produced by the relevant natural causes.
Consideration 2. Thing X is a miracle.
Consideration 3. Thing X is impossible to imitate by me or anyone else for that matter.
One may rightly argue that consideration 3 is logically implied by consideration 1. But the above three considerations aren't premises of an argument, rather they are the chronology of thoughts as they present themselves to someone contemplating a miracle. It would be weird for someone witnessing the parting of the Red Sea, for example, to immediately ask herself: would I be able to do that? That consideration comes a bit later, after the realization that she is witnessing an impossible event. In fact, one could even argue that the consideration of inimitability doesn't arise spontaneously, someone has to point it out. And that's why it's stronger than a "mere" miracle claim, the question of inimitability arises only after one is convinced that X is miracle. Note that an inimitability claim isn't epistemically stronger than a miracle claim, it's stronger in the sense that it has a greater psychological impact.
With the above in mind, it becomes easy to see why the Qur'an argues for its inimitability as opposed to its miracle: it's smart rhetoric. In presenting its arguments- any argument- the Qur’an almost consistently chooses the rhetorically stronger way to do it (e.g. instead of stating “p is true”, the Qur’an would say “isn’t P true?”). Hence inimitability, and not miracle. Also, this formulation is strategically useful to the Qur'an's purposes: since the inimitability claim has the connotation of challenge, the arrogant disbeliever can be intimidated and put to his place by this.
So in apologetics or proselytizing discourses, which term/concept would be favored- inimitability or miracle?
I guess context here is key. In apologetics or academic discussions where rhetoric is to be kept to a minimum, it makes more sense to use the concept of miracle to formulate our arguments. As argued above, inimitability and miracle are epistemically parallel concepts, and miracle ensures easier communication. This is especially the case if inimitability is treated as some irreducible feature of the Qur'an: the proof that the Qur'an is inimitable is that it is inimitable and that's all there is to it, there isn't any feature in the Qur'an which contributes to its inimitability, it's irreducibly inimitable. Consider the following conversation:
Muslim: The Qur'an is from God.
Non-Muslim: What's the proof?
Muslim: The proof is you or no one else can imitate it. Go ahead and try.
I know, arguments like this wouldn't really be common in smart apologetics discourses. But popular usage of the term inimitability might connote things in the vicinity of this. Therefore, arguing that the Qur'an is a miracle would be easier on both sides: the Qur'an is a miracle because it has such-and-such features which cannot be accounted for by the natural causes relevant to the time and place of its revelation. Then it would be easier for both sides to argue for and against this proposition: the Muslim would claim that feature X is indeed beyond natural causes while the non-Muslim would claim the opposite.
Also, it's embarrassing to see Muslims making a strong rhetorical claim without substantiating it (this was how things were played out about Qur'an's literary miracle until very recently). This could be avoided if we stuck to the more modest (rhetorically, not epistemically) miracle claim.
One may argue against this by saying that we are shying away from Islamic Qur'anic vocabulary. But the Qur'an is much more than its vocabulary: it only makes the strong rhetorical claim it does because it provides sufficient evidence which warrants it. Until Muslims can demonstrate such evidence to a substantial epistemic degree, we may want to cut down on the rhetoric.
For proselytizing purposes however, where the point is not necessarily to make a better intellectual case but to convince the prospect (psychologically or intellectually, any way that works), using the inimitability claim is more warranted.