How is a continuous probability distribution different than a discrete one? the probability that it will rain tomorrow?

Continuous distributions define probabilities associated with continuous variables and discrete distributions define probabilities associated with discrete variables.

If a variable can take on any value between 2 specified values it is continuous

otherwise it is discrete.

A discrete variable would be what you mentioned in your question - will it rain tomorrow, or if we flip a coin and count the number of heads that is a discrete variable.

An example of a continuous variable would be the weights of pupils in a school. They could be any weight.