Conscience and conscious come from the same Latin word conscius and their word elements mean with and knowledge, but they are used differently.

conscience: (noun) sense of the quality of one’s character and conduct; adherence to moral principles; consideration of justice.

This is the “voice” in our heads that kicks in when we have a decision to make involving what might be morally right or wrong.

conscious: (adjective) aware or awake; involving thought.

A conscious person is aware of their surroundings, aware of what he or she is doing, and acting deliberately.

It can be used with a noun in a phrasal adjective, as in self-conscious, health conscious, or budget conscious, and refers to someone who is concerned or sensitive about their image, health, or finances, respectively. (antonym) unconscious

Related:

consciousness (noun) mental awareness

conscientious (adjective) scrupulous or careful. A conscientious objector is someone who objects to a requirement on religious grounds, such as vaccinations or serving in the military.