Swiss mercenaries fell out of favor as technological advances rendered them obsolete, and the Battle of Marignano in 1515 during the Italian War is pointed to as an example of their fade from glory.

Swiss mercenaries were considered to be the best of the best of European mercenaries, particularly due to their efficiency using halberds/pikes against cavalry and infantry alike. This is what led to their initial employ by the Vatican and, though they were slowly being replaced by gunpowder units, they were indeed instrumental in the escape of Pope Clement VII during the sack of Rome during a mutiny in the Papal States. I've never really found or heard a concrete reason as to why they stayed on other than tradition, though I would hypothesize that their devoted service is a big part of it. The military wing of The Church was eventually disbanded leaving only the Swiss Guard to protect the life of the Pope and to defend Vatican City.

Nowadays, the Swiss Guard have slowly evolved into a more Secret Service type role. Though they still perform ceremonial duties, such as manning the gatehouses and parading around Vatican City in their flamboyant outfits, they are also increasingly trained to supplement and serve as bodyguards for the Vatican.

Source: Lived in Switzerland, close friends with a Swiss military officer and historian, and an obsession with military history. For intro reading: http://www.swissguard.va/content/guardiasvizzera/en/storia.html

Don't go to the Wikipedia page, by the way. I pulled it up to joggle my memory and found that it cites the websites for the PC games Total War and Civilization. Fun games, but not the best of historical sources.

Edit: As /u/BenBenRodr points out, those are links to the "In popular culture" section. My mistake. Still, though, don't rely on Wikipedia for all your information. The citations page can be a great hunting grounds for further reading, though.