Complicit means being involved in wrongdoing, usually by helping or by knowingly allowing it. Complacent means being comfortably satisfied and insufficiently alert — pleased with how things are when you should be watching. One implies guilt, the other implies inattention.
The core difference
Both are criticisms, but of very different failings.
- complicit — party to wrongdoing: complicit in the cover-up.
- complacent — uncritically satisfied: complacent about security.
Where they meet
Silence can be described either way, which is where the confusion lives. Staying quiet because you benefit is complicit; staying quiet because you assumed everything was fine is complacent. The choice of word makes a specific accusation, so it is worth being deliberate about which one you mean.
Related word
Complicity is the noun for the first. Complaisant — a third similar word — means eager to please, and is now rare enough that using it invites being read as a misspelling of complacent.