Remarque's novel starts with Paul and his company on break from the Western Front, and Kemmerick down and out in the field hospital.
Since the novel's told in first-person narrative by Paul, it can jump around through time to let us know how he and his fellow soldiers wound up serving in the German army.
Given that Milestone's goal with All Quiet was to show the experiences of WWI soldiers as accurately as he could portray them, a first-person narrative wasn't really going to cut it. No voice in the sky informs us about Paul's exploits, because no (sane) soldier goes around narrating his life to an unseen audience.
For the film, the narrative structure was retooled to serve Milestone's needs. We follow Paul and his friends through the events of the war in chronological order. We start with them as young men in school, see them through boot camp, witness their first horrifying days at the Front, and discover them to be old war dogs by the film's conclusion. (Those who survive, at least.)
Interestingly, Milestone doesn't provide any narrative signposts for us to follow. No title card informs us of the date or where the battle's being fought. Milestone chooses not to let his audience know if Paul is fighting in the Battle of Mons or the Battle of Verdun.
If we had to guess why, we'd say it goes back to his desire to show the war from the soldier's perspective. The names and dates of a battle show war through the eyes of a historian—for a soldier, names and dates of battles don't matter at the time. The only thing that matters is surviving the battle at hand, and helping to ensure your fellow soldiers survive as well.