A major difference between Zazu and Polonius is that the former is an escort, working for Simba's mother, of Nala and Simba as they go play. Polonius is no such escort, and he doesn't work for the side Hamlet's on. Rather, Polonius is a spy for the King, Hamlet's enemy, even though Polonius doesn't know that Claudius is a murderer.
Polonius is more of a major character in the play, unlike Zazu, a stock comic relief character. Polonius is complex and round, unlike the flat and static Zazu. Polonius is an active agent in the plot; Zazu merely comments on it. Polonius is a foil for Claudius, King Hamlet, and Old Fortinbras; he is a father whose death helps his children realize themselves.
Polonius' death is a turning point in Hamlet. It is the first time Hamlet has killed, and Polonius' death will give rise to Laertes' revenge, setting up the fated duel at the end. Zazu is not at the center of the action in The Lion King.