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.