I’m usually pretty understanding when it comes to commonly-confused words, but this one annoys me, simply because of how I often I see people get it wrong (almost always using disinterested when they mean uninterested).
OK, so explaining the difference between the two words is pretty straightforward, but why does one begin with dis-, and one with un-?
First, we have to ask: is there a difference between the prefixes dis- and un-? The short answer is no, and the long answer is kind of.
Un- is the older of the two, dating back to Old English, and actually derived from two separate prefixes. First there was on- which was added to verbs to indicate a reversal of the action, and eventually evolved into un-. We can still see this used today in verbs such as untie, undo etc. Then there was un-, the prefix of negation used before adjectives, like unopen, uneven etc.
So two distinct concepts (reversal and negation), but you can see how they came to be conflated, as the negation of a state is effectively the opposite or reverse of that state.
Then a few hundred years later, dis- enters Middle English. Derived from the Latin bis, meaning twice, it was originally used with a sense of separation: disjoin, dismantle etc. Its use exclusively in this manner didn’t last very long though, and it soon became generally indistinguishable from un-. This is probably because a prefix related to separation is quite specific, and people began to use it in more general ways.
And that’s how it is now. Both un- and dis- are both general negative prefixes.
Still, you can kind of see the distinction between the two in uninterested and disinterested. If you’re uninterested in something, you have a simple lack of interest in it. There is no interest. But if you’re disinterested in a situation, you separate yourself from it, and remain neutral.
Well, that’s what I think anyway. Whether you’re interested in the history of the two prefixes or not, please at least try to use disinterested and uninterested correctly.