Every mathematical definition needs to navigate a tradeoff between generality and power. Monoids are more general than groups, but the price you pay for that generality is that it's much harder to say things about monoids than it is to say things about groups. Groups have the isomorphism theorems, Lagrange's theorem, the Sylow theorems, etc. etc.; all good stuff with lots of applications. There are much fewer useful general statements like this that you can make about monoids.
In fact I don't know a single useful general theorem about monoids off the top of my head. Probably they exist but I haven't needed to learn them; I think I've seen Benjamin Steinberg quote a few.
If you want to have some fun you can try working out what the correct substitute for a normal subgroup is for monoids, for the purposes of constructing quotients (hint: it is not a kind of submonoid). This is a nice exercise but also I have never used the fact that I know how to do this to prove anything.