Regime -- We use this word today about a government. The neo-conservatives speak of "regime change."
Regimen -- If you're on a regimen, you're on a set schedule, say of exercise or diet.
Now the problem: The words have a common ancestry. They once were nearly the same. Take a look at Merriam-Webster:
Regime -- 1 a : REGIMEN 1 b : a regular pattern of occurrence or action (as of seasonal rainfall) c : the characteristic behavior or orderly procedure of a natural phenomenon or process
2 a : mode of rule or management b : a form of government (a socialist regime) c : a government in power (predicted that the new regime would fall) d : a period of rule
and then Regimen -- 1 a : a systematic plan (as of diet, therapy, or medication) especially when designed to improve and maintain the health of a patient b : a regular course of action and especially of strenuous training (the daily regimen of a top ballet dancer)
2 : GOVERNMENT, RULE
3 : REGIME 1c
As you can see, the words started out meaning much the same thing. Regime then seems to have diverged, with its latest meanings almost exclusively related to its social studies aspect. Regimen has been consistent for a while, but has been used sparingly to mean regime.
Confused yet? I sure am.
The point is, if you want to be clear to most people, write about Saddam Hussein's regime, not his regimen. That is, unless you're dealing with his aerobics routine.