The difference between apology and regret is important and has appeared in several places recently. For example, in Australia, the Prime Minister, John Howard has refused to apologise to the Aboriginal (or more appropriately Koori people) for the forced removal of Aboriginal children from their families (the so-called stolen generation) up until the 1950s, but he has expressed regret.

In the downing of US surveillance aircraft in China, the US president, George W. Bush has expressed deep regret but has refused to apologise, despite the Chinese government demanding it.

What is the difference? The definition of the the word apology includes an admission of error or wrongdoing on one's part. In the case of John Howard for instance, apologising would mean that the government would be legally liable for compensation to displaced Kooris. In the case of George Bush, it would be an admission of responsibility and hence a loss of face and possibly a liability under international law. Regret merely expresses a desire that the event had not happened, without any acceptance of wrongdoing on one part.

Hence the difference is more like the difference between: "Geez, I wish that hadn't happened" (nice, neutral, no legal consequences) and "That was our fault, we wish it had not happened" (serious implications).