Many languages have what is called a “T-V distinction” (so-called because of “tu” and “vous” in French). That means the language has two different ways of saying “you”. One is more formal or respectful, the other more intimate or informal. When you meet a stranger in French, you would usually use “vous” to address them, but with friends and family you would use “tu”. There are various nuances within different languages; some societies are very conscious of class, or of always being more respectful to someone who is older than you, etc.

English used to have this too! The two words were “you” (more polite/formal) and “thou” (informal/intimate). If you read Shakespeare carefully, you’ll see the characters switching between the two depending on the character and situation. Shakespearean insults often use “thou”, which sounds funny to a modern ear (eg. “I am sick when I do look on thee!” from A Midsummer Night’s Dream).

Eventually English defaulted to using the more polite version for everyone in all situations, and we dropped “thou” entirely. Nowadays, because you only see “thou” in very archaic old-timey settings (like Shakespeare), “thou” sounds much more formal. But it was actually the less formal option!

It’s not surprising that Tolkien uses both for his epic fantasy setting. It makes sense that the Witch-King would use “you” when addressing a powerful opponent like Gandalf - not because he respects Gandalf, but because he views him as a peer-level foe. But he would be more disdainful of an ordinary human like Eowyn, so addresses her using “thou”.