What is the word for an adult who is not mature?

What term can be used for an adult, especially a man, who is in his forties and still behaves like a teenager, shunning responsibilities typical of mature people, preferring to enjoy himself?

I would just call such a person immature, defined by The American Heritage Dictionary as:

adj. Marked by or suggesting a lack of normal maturity: silly, immature behavior.

Perhaps man-child, Peter Pan or perpetual teenager.

I do not think immature is the correct word, as maturity is defined by sociocultural norms. I think the word you're looking for is simply childish, because that describes behavior appropriate for a child (which is generally accepted to be someone who hasn't hit puberty).

Adulthood and Adolescence as States of Mind

What you are describing may be best characterized as someone who is a “late-stage” or “age-inappropriate” adolescent. An adolescent is neither a child nor an adult, but has one foot planted in each of those two worlds. Only once the former gives way to the latter does a person cease to be an adolescent.

That’s because when speaking not legally or biologically, but rather behaviorally and socially, adolescence is a state of mind. It’s a mental condition, a particular way of looking at oneself, one’s fellows, and one’s world.

As a social condition, adolescence persists until the individual lays aside the carefree world of the child. Only then can they take up the mantle of an actual adult with all its concomitant responsibilities.

There is no fixed time for this to occur; it may be early or it may be late, and it may not even be permanent.

Some few do this by age 14 or 15 when they marry, start families, and move into a home of their own, a practice far more common in earlier eras than today in the developed world.
Others delay until their 20s or 30s, or increasingly now in modern society, even into their 40s.
And some there are who forever remain adolescents until they at last die of old age.

A relevant quote from literature

The great American author Gene Wolfe frequently writes of these differences between children and adults, and of that “feet in both camps” world which adolescents for a time inhabit. Perhaps his clearest statement in this area is the following quotation:

Adolescents are simply those people who haven’t as yet chosen between childhood and adulthood. For as long as anyone tries to hold on to the advantages of childhood — the freedom from responsibility, principally — while seeking to lay claim to the best parts of adulthood, such as independence, he is an adolescent. [. . .] Eventually most people choose to be adults, or are forced into it. A very few retreat into childhood and never leave it again. A large number remain adolescents for life.

A person who per your description is “behaving like a teenager” is an adolescent, no matter whether they are 11 years old or 21 — or 51, 71, or 91.

Consider using one of these to describe such behavior: