I guess "clever" is a synonym of "smart"; a kind of intelligence that has nothing to do with one's academic achievements nor one's capacity to resolve ecuations. "Intelligence" is, in my humble opinion, more related to formal intelectuallity, to formal acquired knowledgements. I see "clever" as "sensitive intelligence", wit, the wisdom you gain through experience in your daily life.

I generally agree with other comments here. I think another factor to consider here is usage in American English vs British English. You hear "clever" a lot more in British English, and I believe it generally means intelligent. In the US, clever is used less often, and it seems to imply a little bit of shrewdness / unexpected intuition. If something was cleverly designed, it might imply it was an original and unconventional idea. Smart also means intelligent. However, in British English, smart can also mean put together well, as in "smartly dressed" (you're wearing a nice suit). "You look smart" in American English would imply you look intelligent, as opposed to looking well dressed. (I think that's right...)

Intelligence is the ability to learn, understand and make judgments or have opinions that are based on reason. Clever refers to have or show the ability to learn and understand things quickly and easily. Smart can mean someone who is quick and clever, or just someone who is intelligent.

My take is .... clever can be applied to objects, Example: "Oh, what a clever design!" whereas intelligent is applied to people. You could say "Oh, what an intelligent design", but you would be implying that the designer put a lot of thought into the design. A subtly different emphasis.