"to split, part or divide by force," Old English cleofan, cleven, cliven "to split, separate" (class II strong verb, past tense cleaf, past participle clofen), from Proto-Germanic *kleuban (source also of Old Saxon klioban, Old Norse kljufa, Danish klöve, Dutch kloven, Old High German klioban, German klieben "to cleave, split"), from PIE root *gleubh- "to tear apart, cleave."
Past tense form clave is recorded in Northern writers from 14c. and was used with both verbs (see cleave (v.2)), apparently by analogy with other Middle English strong verbs. Clave was common to c. 1600 and still alive at the time of the KJV; weak past tense cleaved for this verb also emerged in 14c.; cleft is still later. The past participle cloven survives, though mostly in compounds.
"to adhere, cling," Middle English cleven, clevien, cliven, from Old English clifian, cleofian "to stick fast, adhere," also figurative, from West Germanic *klibajan (source also of Old Saxon klibon, Old High German kliban, Dutch kleven, Old High German kleben, German kleben "to stick, cling, adhere"), from PIE *gloi- "to stick" (see clay).
The confusion was less in Old English when cleave (v.1) was a class 2 strong verb; but it has grown since cleave (v.1) weakened, which may be why both are largely superseded by stick (v.) and split (v.).