These are the same word, but gauntlet is the much more common spelling. Gantlet may be seen as an error by many people, although it is nothing more than a variation, albeit much less commonly used.
Gauntlet has two meanings, each a part of a popular English saying. The most common meaning comes from the idiom to run the gauntlet; this refers to an ancient form of punishment, in which a punished person must run between two lines of men holding clubs or sticks, to hit the person running between them. However, in modern speech, “running the gauntlet” can mean any kind of painful challenge, ordeal, or test—a situation in which a person feels attacked while accomplishing a task, one which requires multiple actions. For example:
After running the gauntlet of graduate school exams and her dissertation defense, Selena finally received her Ph.D.
The other meaning of gauntlet is used in the phrase throw down the gauntlet; here, gauntlet refers to an armored glove, as might be worn by a knight in armor, from medieval times. To throw down the gauntlet originally referred to the tradition of one man challenging another to a fight (a duel) by the symbolic act of throwing a glove down in front of his enemy. These days the phrase can be used any time one person challenges another in almost any way, such as in a debate:
The politician threw down the gauntlet when he questioned his opponents qualifications.
Gauntlet is used almost exclusively in these two sayings, although with some flexibility:
She did not feel ready to face the gauntlet of graduate school exams.
To summarize, the spelling gantlet is rarely used, and gauntlet is considered more correct by most people.