There are a number of different relationships between x and y that can be expressed by saying that x is an antonym of y. All of them are covered by the standard dictionary definition of antonym as a term with the opposite meaning, because opposite itself covers a number of different relationships.
(1) If y stands for one end of some kind of a spectrum, then one can say that x is its antonym if it stands for the other end of the same spectrum. If one wants to make it clear that one is talking about a relationship of this kind, one can say that x is the gradable antonym of y. In logic textbooks, the relationship is sometimes expressed by saying that x and y are contrary concepts. A question seeking the gradable antonym of y is meaningless if y does not stand for one end of some kind of a spectrum; most words cannot have gradable antonyms because they do not satisfy that condition. Black is the gradable antonym of white, but there is no gradable antonym of grey, because grey stands for something that is in-between these extremes, and, of course, there is no gradable antonym for desk, rock, or tree because these words stand for something that is not on any kind of a spectrum. It should be noted that spectrum and end are used here loosely and that x and y themselves may be vague; as a result of that, there will sometimes be room for discussion whether x and y are gradable antonyms.
(2) One can, however, also say that x is an antonym of y if x covers everything that is not y, within the relevant domain. If one wants to make it clear that one is talking about a relationship of this kind, one can say that x is a complementary antonym of y. The relationship can also be expressed by saying that x and y are contradictory concepts. One can always create the complementary antonym of a word by adding non-, un- or a similar prefix to it. Not many words, however, have complementary antonyms that are morphologically independent, but some do (e.g. stationary is the complementary antonym of moving).
(3) If x and y are defined as words for the objects in a binary, asymmetrical relation to each other (e.g. pupil and teacher, or buyer and seller), they are also sometimes said to be antonyms, although that use of antonym is probably less frequent than the first two. If one wants to make it clear that one is using antonym in this way, one can say that x is a relational antonym of y.
Usually, somebody seeking an antonym (of any of these kinds) of y expects it to be in the same register as y.
Now, to answer the question directly, the relationship between the concepts of an antonym and negation is the closest in (2), as a complementary antonym of y is, by definition, a word that stands for all and only those things that are not y. Definitions of the other two kinds of antonyms imply negation, but involve more than that: such an antonym stands for some of the things that are not y, but not for all of them.
Questions often appear on this site in which somebody seeks an antonym for a given word without being aware of the differences among different kinds of antonyms, and without realising that the kind of antonym that one has in mind may determine whether it makes sense to ask ‘What is the antonym of y?’ Regular contributors to the site are frustrated with such questions, and it is understandable that they sometimes wish that the word antonym did not exist at all. There is, however, nothing wrong with using that word, as long as one is mindful of the differences among different kinds of antonyms, and makes sure that it is clear which kind is referred to in a particular case.