Finite rhymes with:
ammonite, basnight, bauknight, bentonite, canaanite, clintonite, fortnight, franklinite, goodnight, hignight, hignite, lignite, limonite, maronite, masonite, mennonite, midnight, monzonite, newsnight, reaganite, samsonite, selenite, suburbanite, unite, urbanite, whitenight
Finite sounds like:
faint, fainted, famed, fanned, fant, fanta, feint, femininity, fend, fended, fendi, fendt, fent, fenwood, fiend, find, fined, foment, fomented, fomento, fond, fonda, fondue, font, fonte, found, founded, fuente, fumed, fund, funded