Tact is sensitivity in social situations. A tack is a course or an approach (the word has nautical origins). When switching courses or taking a different approach, one changes tack, not tact.
Tact often appears in place of tack. Presumably some people think of it as short for tactic, which is synonymous with tack in some contexts. This is understandable given how rare tack is, but tact is not conventionally short for tactic, and, fairly or not, phrases like change tact are generally considered wrong by people who pay attention to these things.
The use of tact in place of tack is very common. These are just a few examples out of the many readily available:
Crouch’s fine team goal against AC Milan is testimony to Spurs’ ability to change tact and adopt a more subtle sit. [Sky Sports]
But two companies early on in this proxy season plan to take a different tact. [Fortune]
Fellow Republican Rep. Michele Bachmann of Minnesota took a similar tact as she spoke at the rally. [CNN]
These writers use tack correctly:
He first tried to close loss-making Italian car plants, but changed tack when he saw the backlash that followed. [CTV]
This assertiveness marks a different tack from the way Manila has curried favor with Beijing until now. [Wall Street Journal]
And for good measure, here are two examples of tact used well:
But officers have been ordered to apply the legislation with tact and diplomacy so as not inflame tensions. [Telegraph]
Remember, duck-voiced people with no tact regarding humanitarian crises need not apply. [AV Club]