Yes, they are quite a bit different.
Clever describes someone is intelligent and good at sorting out problems.
They don't necessarily have to be 'smart' to be able to do that. They might just be logical, or come up with a brilliant idea no one else has thought of. eg. A 'clever' solution to the problem where water is pouring through your roof might be to invent a rapid-setting glue which you could just throw onto the roof to stop the leak! That's clever!
Smart has a few different meanings: Someone can be 'smart' if they are neatly dressed. They don't have to be clever, they could be really stupid, but look 'smart'! (eg. some footballers and pop stars for instance).
However, in the context of 'clever', a 'smart' person could describe someone who uses 'street sense' rather than intelligence. eg. if your roof leaks, just move out of the house and sell it to someone who won't notice....that is 'smart' !
A 'smart' person doesn't solve a problem, they work out a way to get round it instead. (that doesn't make them right, or nice people to know though!)