For a long time it was cyclical. The Steelers beat up the Browns for most of the '70s, the Browns got the upper hand in the late '80s, and in the other years the teams would usually split. Pittsburgh's total control of the rivalry actually started in the mid-90s. Belichick lost his last five games against the Steelers as Cleveland's head coach, including the '94 playoff loss.
The thing I really remember about the rivalry growing up was the Browns being unable to beat the Steelers in Three Rivers Stadium. They lost their first 16 games at Three Rivers. They'd change the way they travelled to Pittsburgh, they'd change hotels, once they brought dirt from Municipal Stadium to sprinkle onto the turf at Three Rivers- none of it helped. Of course, the fact that were kept playing the Steelers instead of, say, Duquesne or Carnegie-Mellon probably had a lot to do with their futility in that building.