I am really asking about possessives in general. Do [the World's end] and [the End of the world] have the same meaning. When should you use one over the other? In a related question, why do you use [car of Adam's] instead of [car of Adam] to mean [Adam's car]?
Technically, the end of the world and the world's end mean the same thing. However, the phrase the end of the world in English is typically used to mean an apocalypse. So one typically doesn't use the world's end unless you mean something like the edge of the world. See the first part of LMS's answer for more detail.
In English we vastly prefer the so-called Saxon genitive (not a true genitive, although it used to be) when referring to proper nouns.
We would only rarely use the X of Y when Y is a proper noun, such as when we want to use a non-definite determiner on the possessed object.
That kid of Adam's is a real brat.
Some book of Michael's was lost last week.
A map of Kelly's was on the chart table.
I can only speak as to my dialect (AmE) for the following.
It's important to understand that car of Adam's is correct in this context, and that that car of Adam is incorrect. It's what some people call a double genitive although neither part involved is a true genitive. However, for non-proper nouns, you can use either 's or not. The first is formal.
A crown of the king was stolen.
A crown of the king's was stolen.
If you use the, then the grammaticality also changes:
The crown of the king is beautiful.
The car of Adam is old. (unusual but possible)
In this case, 's is not allowed.
Yes and no. It is possible for "the world's end" and "the end of the world" to have the same meaning, in that they can both refer to the apocalypse or the destruction of the world or society.
However, "world's end" has a second meaning in that it can refer to a remote part of the world, a part of the world so remote that it surely must be where the world stops. Compare with Land's End, the westernmost point in England.
As for the second part of your question ("car of Adam's" vs. "car of Adam"), as far as I am aware the first one isn't correct and is only used informally or colloquially, and the more natural of the forms would be "Adam's car."