What is the difference between a discrete uniform distribution and a continuous uniform distribution?

An uniform distibution is a distribution that gives to all its values the same probability to occur.
Continous case:
the equal probability is given to all the values into an interval ##[a,b]##, which contains an infinite number of values.
the value of the probability will depend on the range of the interval and will be ##f(x)=1/(b-a)##
Discrete case:
the equal probability is given to all the values into a specific set ##S## of values which the distibution can have. A clear example of this is the dice: when you throw a dice you have the set ##S={1,2,3,4,5,6}##, and all the numbers have the same probability to occur (unless someone is cheating 😉 )
the probability will again depende on the cardinality of ##S## and will be ##p(x)=1/n##.