Both "cleverer" and "more clever" are correct. "Clever" is one of those exceptions that English is filled with. Usually the rule is: 1 syllable: add -er (or juse -r if the word ends in 'e' 2 syllables: if it ends in -y, replace -y with -ier; if not, use "more". There are further refinements, but you get the general picture. Other words that are exceptions in the same way as "clever": "quiet", "narrow", "simple"

Indeed, both can be used, although 'cleverer' sounds fugly as hell to me. The double 'er' ("erer") is just awkward. "Quiet" doesn't have that problem. Language is never about applying strict grammar rules, though. You can easily say: "Let's go to a quieter place." The following, however, really sounds forced: "He's cleverer than him." Especially since 'cleverer', when you talk about people, is if not wrong, then certainly not preferred; "He's smarter than him." is much better. "Cleverer" really is only something you say about things, like: "There's got to be cleverer ways of doing this." That sounds perfectly normal again. It's subtle, but distinctly there. And the next, too, sounds very smooth and natural again: "It's simpler than that!"