Between WoD and CoD: Not at all
WoD and CoD are not connected in any canon way. CoD is merely a spiritual successor of the WoD in many regards and at times shares names - which lead to some of the worst Edition Wars ever fought when CoD 1st Edition still claimed to be the (new) World of Darkness and many diehard fans refused to even look at the new products...
In the background, there was a huge rights kerfuffle, White Wolf died & got sold several times, and the Pen&Paper rights ended up at Onyx Path.timeline fuzzy The Time of Judgement books were widely ignored by many fans and they fueled Kickstarters to bring back their old games of the WoD as the 20th century reprint/update. To distinguish the two systems, the (new) World of Darkness got rebranded to Chronicles of Darkness in 2015, a short time after the God Machine Chronicle update to the 2nd Edition... and it's technically retroactive.
The Game Engines
Likewise, the Storytelling system is a spiritual successor of the Storyteller system. On a side note, the Storyteller system is a spiritual successor of Shadowrun, altering its d6 to d10 and removing the exploding dice from normal rolls.
Inside the WoD/CoD
Inside the WoD, all the splats inhabit the same world, and conflict between them is commonly referenced in all the splats books. The scope of the game is much wider: There is a global conspiracy from each splat, the wars are global, conflicts are aeons old and many cornerstones are set deep in the canon of the game. Each Splat comes with either a line it is tied to or it is a fully splat with a core book of its own. Crossplay compatibility is somewhat low, as systems are largely incompatible, but the backdrop of the world is completely shared.
The vast power-level differences though lead to huge fanboy-fights about which was the best Splat to wipe out all the others. Demons and Mummies always won for they came back.
As in cross-referencing, the three main lines (Vampire, Werewolf and Mage) regularly made referencing to one another, and starting 2nd edition even contained stats for these splats when used as adversaries ported to the relative system.
Wraith's end was referenced heavily in Mage revised and became the start point for both Demon the Fallen as well as the last iteration of Mummy: the great Avatar Storm was what destroyed the Wraiths, released the Fallen and came right before the new life spell for the Amite.
Changeling is only passingly noted in Werewolf publications, usually when Arcadia is described.
Expansions into previously nondescript areas or times came somewhat close together: the "Hengeyokai - Shapeshifters of the East" came together with "Kindred of the East", both detailing those splats that are at home in eastern Asia in 1998 under the label of the "Year of the Lotus". They included inter-splat Metaplots.
These Metaplots of the WoD also had often larger impact like the Year of the Reckoning (1999) ending the Wraith line and introducing the Hunter: the reckoning. The Year of the Scarab (2001) was focussing on the Middle East and brought Mummy the Resurrection as a semi-standalone splat (some basic rules had to be taken from either a Vampire, Werewolf or Mage core book). Year of the Damned (2002) introduced the Fallen Demons/Demons. In these plots, the places and plots are detailed from several sides and the stance of the new elements was brought in.
The end result of these Metaplots was meant to be the Time of Judgement (2003/4), which brought the canonical end of the WoD. Following the plot, the WoD is destroyed but how exactly is dependant on the players.1
As the game aged, the 20th-Anniversary edition was made as a fix-up of the game system with a mix of 2nd and revised edition fluff. A lot of edition inconsistencies of the crunch were smoothed out. The general tone stayed, but at times it was modernized for a world that had aged and became considerably more technologically advanced.
The different splats are not really compatible with one another still, as each splat follows vastly different systems for their splat-specific systems, but it has become somewhat less of a headache.
The 5th edition is a rather recent upcoming. It is meant to bring a new system that still has the old roots, but so far only Vampire has made the release and I have not taken an in-depth look at that. So I can just say "it exists and critics are divide about it". It is probably more of a Darker-and-Edgier Reboot.
The basic idea of all splats inhabiting the same world is also true for the CoD, but now the fanboy fights were dulled a lot as the system was much more streamlined. It was designed from the mortal up, not the supernatural down, and tried to even the playing field between Splats. The mechanics for a similar effect are often similar among different splats making cross-splat play easier.
The scope of the game is much narrower: instead of a global conflict with cornerstone figures and an ages old history, the game is set much more local, the backdrop is often fuzzier, and it goes much more to modern punk.
CoD 1e & 2e
CoD 1e and 2e are connected via the God Machine Chronicle plot, which somehow reminds of the Metaplots of the WoD, but is mostly meant to explain the changes to the background between the two editions and to gloss over the mechanics changes.
Do you want to know more?
If you are interested in the history of one specific game, that'd be an excellent history-of-gaming question.
Even with the new products, the WoD stays not connected in game or easily compatible with the CoD but for how it was explained here.
1 - In a nutshell, the games refused to die in the player heads and transitioning to the new World of Darkness (later rebranded Chronicles of Darkness) didn't happen as planned. Some Authors pulled their IP rights from White Wolf and sat on them till they banded together with others and Onyx Path and did the Kickstarter for 20th century.