Why did the two halves of the Roman Empire fall a thousand years apart?

Because by the time the big invasions came, the empire had been split into two halves that were in very different positions. During the Migration Period, large invasions by Germanic peoples and the Huns brought the Western Roman Empire down, and it collapsed in 476. In the East, the Theodosian Walls of Constantinople stopped Attila, the rich provinces of Anatolia, the Levant and Egypt escaped any significant invasion from across the sea, and their taxes kept paying for the essential work of the state. So the Eastern half, better known as the Byzantine Empire, kept going for another thousand years, until Constantinople fell in 1453.

The empire itself was born from crisis. By 100 BC Rome already ruled most of the Mediterranean, but civil wars destabilised the Republic until Octavian defeated Mark Antony and Cleopatra at Actium in 31 BC. In 27 BC the Senate gave him sweeping military power and a new title, Augustus, making him the first emperor. Two centuries of stability followed, the Pax Romana, and the empire reached its largest size under Trajan before trouble began under Commodus.

The split grew out of another crisis. In the 3rd century, fifty years of civil war, plague and invasion threatened the empire's existence, and whole regions broke away before Aurelian reunified it. Diocletian, who ended the civil wars, then set up two imperial courts, one in the Greek-speaking East and one in the Latin-speaking West. In 330 Constantine the Great, the first Christian emperor, moved the imperial seat from Rome to Byzantium and renamed it Constantinople.

Because the empire lasted so long and reached so far, its institutions and culture left a lasting mark on language, religion, art, architecture, law and government across its former lands.