Why didn't Christianity spread east the way it did to the west?

There were actually significant Christian communities (called the Nestorians) in the Sasanian Empire, mainly concentrated in the foothills of the Zagros in northern Iraq and around the Persian Gulf, and they were quite active, holding church councils that met in the imperial capital of Ctesiphon that presided over a flock that stretched as far east as Zerang and Qandahar in modern Afghanistan, and Merv in modern Turkmenistan.

It's also important to remember that even under the Sasanians, the Christian communities were never integrated into the the state cult as it was in the late Roman Empire. Most of the aristocracy adhered to the traditional Zoroastrian faith, so while it was a flourishing community, it was not as ingrained into the avenues of power and prestige that persisted even after the physical collapse of Rome, whose successors often adopted the trappings, symbols, and faiths of imperial Roman authority (see /u/telkanuru below for more detail). In the East, Islam became the religion of power and prestige and those of different confessions either slowly converted, left, or remained as a minority. We actually have documentation of Nestorian communities, fleeing the Sasanian collapse, that arrived in China during the mid-seventh century!

TL;DR There were flourishing Christian communities in the sixth century Sasanian Empire, but even then it was never the religion of the aristocracy, so that with the arrival of Arab Empire and the Caliphs, which had its own networks of religious authority and prestige, Islam became the source of elite cultural status, sidelining not only the Nestorians, but the Zoroastrians as well. Over time, Christian and other religious communities either converted, left, or remained as a small minority.

EDIT: poor syntax

EDIT2: Removed assumption in chronology