I know that the light novel Fate/Apocrypha has some divergent point after the Third Grail War, but I was wondering what exactly that divergent point was. How did the world end up without a Fourth and Fifth Grail war? Did someone win the Third Grail War, or was there something else that happened differently?
One cause was Darnic Prestone Yggdmillennia, who aided the Nazi's during the Third Holy Grail War by locating the greater Grail and stealing it.
Seventy years ago, he participated in Fuyuki's Third Holy Grail War in an unknown capacity, aiding the Nazis to supposedly help them further their own goals. He found the location of the Greater Grail, and with the power of the Nazi army behind him, plundered it. He betrayed the Nazis as it was being transported to Germany without their knowledge, leading to the only details being known that the Grail disappeared without a trace and the magus aiding them also vanished.
However, there is also another deviance which dates back earlier than Darnic.
In the original timeline that Fate/Zero and Fate/Stay Night follow from, the Einzberns summoned a new Avenger Class for Angra Mainyu. However, its early defeat and storage in the Greater Grail corrupted it, causing the abnormalities we see in future wars (non-Heroic Spirits being summoned, wishes becoming destruction).
In Fate/Apocrypha however, the Einzberns summoned Ruler, which in the Great Holy Grail War is Joan of Arc, but in the Third War, it was Shirou Tokisada Amakusa who we know in Apocrypha as Shirou Kotomine
As the third war drew to a close, the Einzberns clearly stood closer than any other to the Greater Grail. However Darnic Prestone Yggdmillennia successfully stolen the Greater Grail, the only survivors in Fuyuki City were Shirou and Risei Kotomine. While his Master was killed in action, Shirou remained materialized due to Shirou came in contact with the Greater Grail and he managed to receive flesh.
So we can assume that what caused the timeline to split is when the Einzberns chose to Summon Ruler instead of Avenger.