Firstly, the grammar: Grammatically, it's the same as the difference between: I go nowhere I don't go anywhere I know nobody I don't know anybody I understand nothing I don't understand anything In the first sentence of each pair, the negative element is in the object (nowhere/nobody/nothing/ no difference), while in the second sentence the negative element is in the verb (don't ....) . Secondly, the meaning: Simultaneously giving you both another example and an explanation: There is no difference in meaning between these sentences. There isn't any difference in meaning between these sentences. ! Thirdly, the use: The second form (negative verb + any) is the more common. The first form is more emphatic, and generally used less. For example, you wouldn't normally say 'I drank no coffee' - you'd say 'I didn't drink any coffee'. The 'no coffee' form is grammatically correct, but sounds quite unnatural. Going back to your original question, I'd say that 'I see no difference' is a kind of 'set phrase'. The construction is a little unusual, but the phrase itself is quite common.

Like Michael said, you would need an "any" after "do not see." However, even though there are no differences in meaning, "see no" is, to me, very formal and literary. It's rarely heard in informal, everyday conversation, so I suggest not using it unless you want to sound scholarly or academic. I recommend using the much more natural-sounding "do not/don't see" more frequently than "see no." ^.^

You'll need a partitive article ("any") but then, to my US/Canadian English mind they're the same. I see no difference = I do not see any difference