Firstly, Hiccup is Clojure (so basically server only). Cloact is ClojureScript, that compiles to javascript (so browser, node.js, etc).
Secondly, Hiccup produces strings. Cloact typically creates DOM elements, through React, that change every time a component rendered. But it can also produce an HTML string using the same data, to pre-render a page on the server using node.js.
The React.js system then is responsible for pushing it into the DOM, which it can often do in a way that's more performant than using a string intermediary (by computing deltas)
That's the short of it- The other part is that React is mainly a client-side technology whereas Hiccup is mainly used on the server (though these lines are blurry)
As danh and drcode said, Hiccup is basically edn->text and server side only but there are a number of implementations of the same concept in ClojureScript:
