Intelligence is an ability which you are more or less born with – it is inherent, and determined by your genetic makeup. Your IQ, for example is a measure of your intelligence, and your ability and capacity to learn things. On the other hand, “smart” can often mean learned inferences – e.g. being “street-smart” or “book-smart”. When you learn, study or practice something repeatedly, you become smarter on the subject. In this sense, intelligence is inherited, while “smart” is an earned, acquired status.
Intelligence refers to a higher degree of intellect, and when someone is praised as “intelligent”, the compliment is far higher than merely referring to someone as “smart”. Calling someone “smart” can also have negative connotations, as it is often used as a substitute for sarcastic – e.g. a “smart-alec” answer – or to refer to someone who has a sharp tongue. However, the word “intelligent” can never be perceived as negative in any sense.
Smart can also be used to describe someone’s appearance. For example, if you happen to be wearing a new tie, a particularly sharp suit, or a well-tailored outfit, you are likely to be described as looking “smart”. This means that you look good in the particular circumstances, and present yourself well – it is by no means a measure of mental ability – e.g. there is no such thing as an “intelligent dresser”. Smart can be used in more superficial terms, while “intelligent” is always an adjective that carries more weight.
Intelligence is often measured by a person’s education – completing a certain degree of higher education makes it more likely for a certain person to be referred to as “intelligent”. However, “smart” is hardly ever used for individuals who have completed their higher education – you might be “smart” if you managed to skip a grade during elementary school, but will be undoubtedly “intelligent” if you recently submitted your PhD thesis.