Said to me today: “I don’t want that on my conscious.”
Me: “You probably don’t want it on your conscience, either.” 
Them: “Huh?”
Me: “They are different words.” 
Them: “Really?”
Me: “I promise you.”

These commonly confused words sound similar but they are not homophones. 

Conscious is an adjective. It is a descriptive word that means awake or aware. 

Examples:
I’m conscious of the confusion between words that sound similar but which are very different in meaning.
He passed out, but he is conscious again now. 

Conscience is a noun. It’s the name given to that part of our being that tells us not to do something we know we shouldn’t, and accuses us when we have done something wrong so that we feel bad about it. 

Examples:
She was good at acting innocent, but her conscience was plagued by guilt.
His conscience reminded him daily of the things he had done. 

The difference in the way these words sound is minor, but the difference in meaning is significant.