Only by social convention. Not enforced by the language's semantics in any meaningful way. In fact, Java does not even allow the programmer to define custom values. All values are primitives or object identities. Note that so-called “value objects” will not fix anything.
A fundamental property of functions is so-called “function extensionality”. Two functions “f, g : Foo -> Bar” are equal if and only if “f(x) = g(x)” for every “x \in Foo”. In particular, this means that there should be no equality testing operator for functions, because function equality is undecidable.
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OCaml at least has bona fide values. It allows you to implement procedures that evaluate interesting functions (i.e. value-to-value mappings), even if it doesn't typefully distinguish a special class of procedures that are syntactically guaranteed to evaluate functions.
F# and Clojure are lost cases, though.
As for Haskell's IO and IORef (or ML's ref), there is absolutely nothing wrong with the ability to implement procedures that do other things besides evaluating functions. The problem is when you lack the ability to express interesting functions, and the root cause is the lack of a sufficiently rich universe of values.
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Your mention of purity completely misses the point. The ability to do traditional imperative programming is a feature, not a bug! The problem here is the inability to do functional programming. The reasons for this are technical, not cultural.