Tack is a course or direction, from sailing — a boat changes tack by turning across the wind. Tact is sensitivity in handling people. The phrase is change tack, not change tact, and "take a different tack" is likewise nautical rather than diplomatic.
The core difference
A sailing term and a social skill, joined only by sound.
- tack — a course or direction: try a different tack. Also a small nail, and to fasten.
- tact — sensitivity, discretion: handled with tact.
Where the phrase comes from
A sailing vessel cannot head straight into the wind, so it advances by zigzagging — each leg is a tack. Changing tack means altering approach to make progress, which is exactly the figurative sense. Understanding the sailing image makes the spelling obvious.
The related words
Tactful and tactless come from tact. Tactic and tactical look related to both but come from Greek taktikos, concerning arrangement — a third, separate root, which is part of why the pair gets muddled.