My question is that I know that there is an old and a new WoD. Old WoD ended with the apocalypse. And there is the new WoD, which as far as I known does not contain references to an apocalypse.
Does that mean, that when playing the new WoD we should treat everything else as never happened, as a completely different world? Or is there some kind of connection between the two worlds?
Despite the many similarities, they are different games with different systems and settings.
nWoD is a reimagined oWoD. The relationship between the two is much like that between 1970s Battlestar Galactica and 2000s Battlestar Galactica. Some core concepts have been kept, others have been significantly altered while reusing names, and some elements are entirely new.
nWoD was intended to replace oWod entirely, but demand for oWoD (and a new publisher) caused the line to be revived.
Now both versions of the setting are ongoing concerns, the publisher has rebranded them to become "Classic World of Darkness" and "Chronicles of Darkness" to further highlight the differences. The name change is being applied at roughly the same time as new editions of the core books are being produced (a takeover of the copyright holder has provided a new owner who is allowing the licensee to issue 2nd editions).
So yes, treat them as completely separate entities.
As already answered, they are completely different games with different rules and different plot. But one should not be afraid of learning both and importing concepts. There is even a set of books about translating features among those two games, one book per setting.
There are many conceptual differences between those games that may make translations harder, which are described in those books.
