Gantlet was the original spelling of the word referring to a form of punishment in which people armed with sticks or other weapons arrange themselves in two lines and beat a person forced to run between them. It came from the earlier English word gantlope, which in turn comes from the Swedish gatlopp.1 Gauntlet is an alternative spelling of gantlet, but it also has several definitions of its own, mostly related to gloves.
Gantlet was the preferred spelling in early use of the phrase run the gauntlet—meaning to suffer punishment by gantlet or to endure an onslaught or ordeal—but gauntlet prevailed by the 18th century. Today, most writers use gauntlet, though gantlet, which is especially common in American English, is not incorrect.
The phrase throw down the gauntlet, meaning to issue or accept a challenge, uses gauntlet in its glove-related sense. It derives from the practice among medieval knights of challenging each other to duels by throwing down their gauntlets. So gantlet does not work as an alternative spelling here.