Question

Difference between discrete and continuous charge distribution?

Answer 1

Discrete systems are those systems in which are made up of finite component particles a which are non-homogeneously arranged such that no smooth variation exists. It is such that all constituent particles have properties which vary randomly. They are direct opposite to continuous systems, which are smooth arrangement of particles which cannot be individually taken into consideration.
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Answer 2

Interesting question. I think it's a question of scale. After all, charge is "discrete" or "quantised" into electronic charges.
So, relatively few of them - eg in an atom - would be discrete.
A lot of them - eg in an electric current - would be "continuous" (or an avalanche of discretes = a continuum ... )