What's the difference between "regime" and "regimen"?

Generally speaking, the two words have the same ultimate etymology, from Latin regimin, meaning “position of authority, direction, set of rules”. In many cases, either word can be used, and their meanings have substantial overlap:

regime
1 a : regimen¹b : 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 d : a period of rule

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 athletes> 2 : government, rule 3 : regime ^1c

The definitions reference each other in several places reflecting the substantial overlap in the two words’ meanings. However, the two words are frequently used differently, so let’s look at what collocates (frequent neighbor words) using the Corpus of Contemporary American English (COCA):

These words most commonly appear before regimen (function words like the excluded)

EXERCISE   116
TRAINING   109
TREATMENT   80
DAILY       65
WORKOUT     39
DRUG        38
FITNESS     36
STRICT      33
MEDICAL     24

Whereas these words commonly appear before regime:

MILITARY       449
COMMUNIST      383
IRAQI          232
AUTHORITARIAN  203
OLD            200
DEMOCRATIC     156
TALIBAN        145
CASTRO         139
NAZI           139

From these results we can see clearly that sense 2 of regime, the one having to do with government, is the most salient one for that word (at least when it is used with an attributive). However, if we look up the frequent collocates for regimen with regime, we do find some (albeit fewer) results:

EXERCISE   20
TRAINING    6
TREATMENT   9
DAILY       4
WORKOUT     6
DRUG        6
FITNESS     4
STRICT      8
MEDICAL     1

From this we can conclude that indeed the sense of regime meaning “a systematic plan” or “regular course of action” is also a current usage, and not one that merits criticism for being a “misuse”.

In US English,

• 'a regime change' is, for example, a coup/junta/putsch in a government

• 'a regimen change' is, for example, eating more yogurt.

'Regimen' is all but unheard of in Australian English either. We use 'regime', whether it's exercise, diet, military or government contexts.