There's no way to answer this with any great confidence.
As @Rugnir has noted, the spherical moon and earth were well known already to the Greeks, and there is good reason to think that this was also known in early China and India. If memory serves, it is somewhat unclear whether the ancient Mesopotamians knew this. But in any event, a rotating moon would presumably not speed up this early discovery.
One thing is that the analogy between a rotating moon and the rotating earth might have led somewhat earlier to a strong preference for heliocentric models. These were in fact proposed quite early among the Greeks, but were so annoying to calculate that, in the absence of strong evidence in their favor, they tended not to be followed up effectively. Since at base Copernicus's proposal was a matter of advanced geometry rather than (for instance) algebra, it could have been developed earlier. And if there were a strong preference for such a model, it might well have stuck firmly enough for geometers and astronomers to work out the kinks (as did not happen in European history until relatively late in the 16th century and on into the 17th with Kepler).
In addition, because the moon rotates and thus alters its appearance, the use of lenses to examine its changing faces might have been of significant interest rather earlier than Galileo Galilei.
On the same point, the irregular image of the moon might have led to a much earlier undermining of the "perfection in the heavens" assumption that dominates Ptolemaic astronomy.
Cultures do tend to come up with notions about what the moon looks like, and develop mythological conceptions and descriptions. In Japan and parts of the Americas, the moon looks like a rabbit -- in Japan, it's a rabbit pounding mochi rice. In Europe, an old man's face is more common, though there are of course other notions.
I suggest that the changing faces of the moon would be classified and named, on the model of constellations. Within any given culture, a "rabbit moon" would differ from a "scorpion moon" or a "flower moon," or whatever. Each would then generate a range of mythological conceptions and stories.
These might in some cases have relatively concrete effects. For instance, it is quite common to ascribe both vulpine (wolf-like) and insane qualities to a full moon (thus luna-tics and werewolves). With a rotating moon, you might well get these kinds of ideas ascribed to particular faces of the moon rather than its phases.
More interestingly, I suspect you'd get both: a scorpion moon at the waxing half is understood to be especially dangerous, a full flower moon is especially calming, and so forth.
If I were doing this for a story, I'd probably come up with a bunch of images and concepts for the faces, then work out their notional implications at especially strong phases. But I don't know if you can get decent images of what the moon would look like -- of course, you don't have to use our moon.
Note that in many ways the moon will take on a considerably stronger role in astrological thought, but whether that's of use to you depends on whether you're planning to work out anything much in the way of astrology in the first place.