For many, the Pope is an authority figure of major importance. For people who find comfort in looking to and following an authority, having no pope is like having no boss, or no teacher. When the new boss or teacher finally arrives, there is a sense of relief. More broadly, the Pope is a living symbol of the Catholic Church, in this sense much like the British monarch. Filling the vacancy is like removing the scaffolding from the Statue of Liberty. The symbol is available again.
In practical terms, the Pope is much more than the Bishop of Rome, or a figurehead. He is the ultimate authority for all things Catholic: everything from who gets to be a priest, to the church position on things like birth control and homosexuality, to how the Mass is celebrated. When speaking officially on such a matter ("ex cathedra") the Pope is considered infallible. In this sense, he is more like the lead Ayatollah in Iran, though his secular authority is limited to Vatican City.