There are 3 definitions for "adult", and they are based on different points of view.
Definition in the law: Anyway who is 18 years or older.
This is merely a legal mark. At this mark you get the right to vote, your signature has legal value, and you are responsible for your own money etc. This is far from ideal, but since it's undoable for the law to make sure who is mature enough or not, they had to draw a line somewhere, and 18 was the chosen age.
Biological definition: Anyone whose reproductions organs have begin to fully function.
In other words for boys being an adult means you can make a girl pregnant, and for girls it means she can get pregnant. In the entire animal kingdom the gaining the ability to reproduce yourself means you have biologically become an adult. (There are a few exceptions as to some people this never actually gets to work at all for several reasons. But under normal circumstances, that definition goes).
Social/mental definition: Well, no need for me to repeat that, as the meme already does that well.
Since the three points of view have all different criteria I guess it's obvious why somebody who is physically an adult an still act like a kid and vice versa. Now I do not know about other countries, but the Dutch law has a status called "curatele" in which you do lose all your privileges as an adult, and a curator will take this over from you (like your parents would do when you are still a kid). The curator will then make sure your bills are paid, and your curator has to sign stuff for you. This status can only be applied when people think you cannot handle being the privileges of being an adult properly bringing yourself in trouble. In court a judge has to make sure if the claim is true. Some people have this status on their own request. That too is possible. Being physically and socially mature does not mean you can take all responsibilities an adult has. The "curatele" status provides in the void this can cause.