The noun whit has one definition: the smallest particle. It is usually used in the phrase not a whit, which means not at all or not one iota. Wit has several meanings, including (1) intelligence, (2) cleverness, and (3) a person with wit. It is the correct word in the idioms at wits’ end and to wit. Not a wit, a common error, doesn’t make logical sense (unless it’s used to insult someone’s cleverness).
As for me, I don’t feel a whit less hawkish than I did right after 9/11. [National Review Online]
Most of the time, the president of a sports franchise matters not a whit to the fan base. [Hamilton Spectator]
It matters not a whit if one agrees or disagrees with Palin on issues. [Irish Times]