Apologize is the preferred spelling in American and Canadian English, while apologise is preferred virtually everywhere else. Other than the spelling, there is no difference between the words.
Examples
For example, these North American publications prefer apologize:
Nearly a third spontaneously apologized for their crimes and showed signs of true repentance . . . [Toronto Star]
The fine is to be even higher if Chevron doesn’t publicly apologize. [Wall Street Journal]
Executives from a Japanese Sony music unit apologized Wednesday for a rock band under its management that dressed up like Nazis on a national TV broadcast. [Winnipeg Free Press]
British discount carrier easyJet is apologizing to Jewish customers after it stocked only bacon baguettes and ham melts as meal selections on a flight from Tel Aviv earlier this month. [USA Today]
And these Australian and British publications always use apologise:
I am not ashamed to say that I was too flabbergasted to apologise. [Telegraph]
It is not the first time Ms Ferguson has had to apologise for her financial dealings. [Herald Sun]
Clare this week apologised to Jessica for any hurt. [Guardian]
