A complete market is one in which a replicating startegy exists for any derivative. we just assume that the market for securities is complete (without proof) and hence the derivation of the Black-Scholes formula works.
A perfect market is perfect is the sense of perfect competition as discussed in the CT7 and include a very large number of rational and well-informed investors, none of whom is able to influence the market prices of securities, plus no taxes and transactions costs.