These words seem so similar because they both entered the language from the same root word, but they entered at different times. (source: dictionary.com)
Pliant came into Middle English from Old French between 1300-1350. At that time the modern word "ply" also came into the language as "plien" (from the Old French word "pleier"). The present participle form of "plien" was apparently "pliant" which was the root "plien/ply" + the Latin suffix "-ant" meaning “characterized by or serving in the capacity of”.
Then, with pliant already in the language as the present participle form of plien(ply), pliable separately came into late Middle English from modern French between 1425-1475. Modern French used "plier" (and that change in the French word apparently eventually helped change "plien" in Middle English to the modern "ply"). From the French "pli(er) + the Latin suffix "-able" meaning “capable of, susceptible of, fit for, tending to, given to,” came the adjective "pliable".
As the language moved into modern English, the present participle of ply became plying but it seems the old Middle English present participle form pliant was popular enough that it survived by becoming an adjective, just like pliable. Thus both words are fundamentally the same, both adjectives of the same root word ply and with Latin suffixes that are so close that there is no fundamental difference. And since modern English they have basically been used almost interchangeably.