The use of "nude" is virtually always to indicate that a human form lacks clothing. It is, in most usages, a statement of this simple truth, and so it generally has a more innocent, natural connotation. Because it refers specifically to human forms and cannot usually be used to describe anything else, it can be used as a noun: "a nude" is a person who is nude. A "nude beach" is an example of the word in this noun form; the beach itself isn't nude, it's instead a beach for nudes.
Again, because of the simplicity implied in the word, "nude" has an artistic connotation, as countless works of art depict the human form lacking clothing (at least in areas of the body that we typically conceal in Western culture), without the subjects seeming to be aware of this fact. They certainly do not draw attention to it themselves, although the artist might in the composition of the work.
"Naked" has the same technical definition of being "uncovered", but various idiomatic usages typically give the word a connotation of starkness; something "naked" has often been actively uncovered or exposed, when it is normally concealed; e.g. "the naked truth". This starkness implies an impropriety to the state; that something which is "naked" shouldn't be, because it normally isn't, and the intent of it being naked is to shock or embarrass; it's right there, in your face, daring you to do something about it. Other usages are clinical or technical; "the naked eye". The combination of these various usages give "naked" a "cold" connotation.
A third synonym, with similar general-purpose usage and starkness as "naked" but with a less negative connotation, is "bare". Again, the technical definition is basically "uncovered, unconcealed, exposed", but unlike "nude" or even "naked", almost anything can be bare. The connotation, then, is that it's not necessarily improper for something to be bare; it's more acceptable, however still somewhat stark, and the connotation is that this might not be a good long-term state of affairs for whatever is bare, but it's acceptable, at least for a time. For instance, bare (unfinished) wood.