There are two terms used for pairs of words (in the same or different languages) that look similar but are actually unrelated: false friend and false cognate. Are these terms synonymous? If not, what's the difference?
They are distinct.
True cognates, true friends
Words with a common etymology and modern form and meaning. These are extremely numerous, of course.
True cognates, false friends
Words with a common etymology but which have shifted to have different meanings.
False cognates, true friends
Words etymologically unrelated that nevertheless now have similar meanings (typically by coincidence, but sometimes by influence/reinforcement).
False cognates, false friends
These are completely unrelated words that happen to look similar:
Most often, the expression “false cognate” is used as a synonym for “false friend”. If you google with them, you will mostly find pages that use them synonymously.
However, other meanings have also been proposed. In Concise Encyclopedia of Semantics edited by Keith Allan, the article “False friends”, p. 308–309, describes false cognates as a special case of false friends, namely false friends that are not etymologically related. And in a Unilang discussion, it was suggested (without references) that “false cognates are words that look the same and have the same meaning but have different roots”, so that false cognates would not be false friends at all.
Thus, the expression “false cognate” is best avoided, since it has different meanings to different people. When you encounter it, assume that it probably means “false friend”, but with some suspicion: it might mean something rather different.