Unless you want to get into the specifics (which would be a REALLY long thread), in general From the Ashes takes place just after the Greyhawk Wars. This is a dark time for the Flanaess, with the rise of Iuz's empire, the Great Kingdom's expansion west, and the Scarlet Brotherhood's intrigues. The goodly kingdoms are either in ruins (Shield Lands), economically strained (Nyrond), or politically exhausted (Furyondy).
The Adventure Begins was a last ditch reboot by TSR, late in the 2E cycle, because they realized that a lot of players really hated the "dark" Greyhawk. It setup events that reversed much of the events of the Greyhawk Wars, limiting the power of the evil nations, and allowing the good ones to recover some. There was more intrigue (especially with Nyrond's royal family, IIRC), and less overall warfare.
By the time Living Greyhawk came about, the region was tenuously stable. Living Greyhawk also went into detail on areas, gods, and people that were previously uncharted. It details out every living deity specific to Greyhawk, giving each at least half a page. Unless you are looking for Gygaxian style Greyhawk, Living Greyhawk is probably the best overall single sourcebook for Greyhawk. Unfortunately, it's as dry as an encyclopedia, which is what it really is.