These concepts are distinct, but all too often, they're conflated. What's more, each of the three terms "serious", "formal", and "mature" has multiple meanings which can themselves be blended in misleading ways.
"Serious" can mean "grave", "sullen", "humorless" on the one hand, or "dedicated", "purposeful", "deliberate" on the other. Clearly, one can be simultaneously lighthearted and focused, like a funny math teacher, or solemn and capricious, like a severely depressed person. "Formal" means to satisfy high standards; whether those standards are etiquette or logical rigor depends on the context. You could speak precisely and colloquially (as I try to do) or you could speak vaguely with elevated diction (as politicians often do), and either way you'd be simultaneously formal and informal.
No one denies that cases like these exist, and yet people often behave as if these four separate meanings were tightly bound to each other. Too many employers want candidates for a job to show up wearing fancy clothes, as if only serious thinkers could dress formally. Too many math textbooks are written in a dry, stilted style, as if conventional formalities and a serious tone would help one seriously understand formal methods.
Maturity is a more extreme instance of the same phenomenon. Literally, something that's "mature" is something that's undergone a process of development. In this sense, an organism or a computer program can be mature. Metaphorically, people call something "mature" to mean that it has to do with adults as opposed to children.* So the ESRB people rate the GTA games as Mature to mean that they want to keep children from playing those games, and people use "emotional maturity" to mean "emotional stability" because they believe that children are more neurotic than adults. Obviously, the world has no shortage of "mature" children and "immature" adults, for any kind of metaphorical maturity you can think of.
So each of these three terms has two meanings of interest here. Now notice that the six meanings may be grouped into two themes: solemnity, etiquette, and age are superficial qualities, in that they don't generally tell us how to value a person or behavior, whereas dedication, rigor, and responsibility are important and useful qualities. By confusing these concepts, then, we're thinking the shallow deep and the deep shallow. I don't think that the ambiguity in the words "serious", "formal", and "mature" creates this confusion so much as reflects and slightly worsens it. Perhaps the real issue is as simple as judging books by their covers, which problem would take a lot more than language reform to ameliorate.
Now that I've written this essay, it seems as if it belabors the obvious. Oh well, maybe somebody will find it interesting.
* To head off another potential source of ambiguity, note that I mean the word "children" to include anybody under the age of eighteen, not thirteen or something.
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"If I pick nine, you'd better not be racist against me!"
"No promises, niney."