I can’t count the number of times I’ve had someone tell me they caught a typo I missed: I wrote, “take a different tack” when I must have meant “take a different tact.” I’ll admit I sometimes miss typos, but that’s not one of them. It’s possibly the most widely misused phrase I can think of.
“Tack” — the correct word in this context — is actually derived from sailing terminology. The tack is the lower leading corner of the sail; it points the direction the ship is heading. So when a sailboat changes course, it’s changing from one tack to another, or “taking a different tack.”
Tact, on the other hand, really only has one meaning. It’s a keen perception of what is appropriate or considerate. (Think of tactile–>touch–>the right touch.)