A regimen is a systematic approach to diet, medicine, or exercise. The word has other meanings, but this is its most common use. Regime can carry the same meaning, but it has additional definitions—mainly, (1) a form of government, and (2) a government in power—that it doesn’t share with regimen. Keeping the words separate might be a good idea for clarity’s sake, but in practice they are both commonly used in reference to systematic approaches to things.
British writers are more likely than American writers to use regime for both purposes, but that doesn’t mean that regimen is an especially American word. Regimen in this sense in fact predates this use of regime and is several centuries older than the United States.1 Regimen is currently faddish in the U.S. and out of style in the U.K., but there have been periods of history in which it was far more common than regime even in British writing.
In real-world usage, regimen is most often used to denote a regulated system, as of diet or fitness—for example:
But changing your skin-care regimen between Halloween and Memorial Day can bring about a happier face “pretty quickly,” she says. [USA Today]
In the 1920s, only six nuns still inhabit the convent, contentedly following a regimen of solitude and contemplation. [Independent]
He practices yoga, part of a fitness program that is coupled with an aggressive drug regimen. [New York Times]
And regime is more often used to refer to a form of government or a particular administration—for example:
The entire eastern region and parts of western Libya near the border with Tunisia have already slipped from the regime’s grip. [Financial Times]
It’s been a hard winter in North Korea, and the Kim family regime is once again struggling to feed its people. [Wall Street Journal]
Already, 5,000 Tunisians have attempted to migrate to Italy since the collapse of the Ben Ali regime. [National Post]