Modern English is made up of pieces of different languages (like making a car in a junkyard from parts of different cars). Mostly it's related to German and Latin (through the Norman conquest by the french). So many times there are two words that mean very similar things but one is a shorter one with a German root (foe) and one is longer with a Latin root (enemy).

'Foe' is just a formal way to say 'enemy'. I like fionagibson's example from Shakespeare. It is usually in old, written texts that we see 'foe'. The first time I heard 'foe' was in the question, "Who goes there? Friend or foe?". "Who goes there?" is a literary/old-fashioned way of saying "Who are you? Who is there?" (very often when someone else is nearby but you cannot see them/their face, and you want to know who they are). In modern day, someone might say this as a joke.

"Enemy" is the basic, normal, common, everyday word. "Foe" is old-fashioned and literary. It is, however, part of the everyday expression "friend or foe?" The word "adversary" is formal, technical, and a little softer than "enemy." It is diplomatic language. It does not necessarily carry the idea of war and killing. In a court of law, the lawyers on opposing sides are not really enemies. They would not shoot each other! But they are "adversaries," they are struggling against each other on opposite sides. The word "rivals" means two people (or companies or nations) who are competing against each other for the same thing. The two teams in a football game are not enemies, not foes, not adversaries--but they are rivals. The United States and China are economic rivals. Two women who are trying to marry the same man can be called "rivals." "Foe" is often used in English poetry because it is a short, strong, one-syllable word that rhymes easily: "I was angry with my friend; I told my wrath, my wrath did end. I was angry with my foe: I told it not, my wrath did grow." That's from a poem by William Blake, 1757–1827. It would be very hard to fit the words "enemy" (or "rival" or "adversary") into this poem!

Competitors in an athletic competition are sometimes referred to as "foes". They are not enemies. An enemy is something far more serious. Enemies are always foes, but foes are not always enemies.

I agree with the poster above. I would add that “foe” is a word not really used in modern English in the same way that “enemy” might be. For example, enemies from a Shakespeare play would be “foes”. “From forth the fatal loins of these two foes…” from Romeo and Juliet.Foe is more of an early modern English usage, even though it’s technically correct.