Smartness is just a measure of what a person knows. A person who knows a lot of things (Hiruzen, Black Zetsu, Orochimaru, Kabuto, Madara) is "smart" only because he/she has taken the time to gather and retain information.
Intelligence is how one applies what they know. Simply knowing things can't help a person who doesn't have the ability to apply what they know to a real problem. Innovators (Tobirama, Minato) and people with deductive reasoning skills (Itachi, Kakashi, Shikamaru, Shikaku) are intelligent because they are able to apply what they know to solve an unfamiliar problem or to figure out how to create something new.
I would relate it to math and physics. Math is smarts. Someone who is great at math has knowledge of theorems and formulae necessary to solve made-up problems. Physics is the application of math. It requires one to truly understand what these mathematical theorems and formulae actually mean and how they can apply to the real world. People who are great at math are not necessarily good at physics, and people who aren't too good at math can be very good at physics, simply because they understand the math that they are doing, even if they don't know as much about math as others in general.
The important thing here is that intelligence can't be taught. Anyone can be smart though.