In a Poisson process, the times between successive events are described by independent identical Exponential distributions. In a renewal process, like a Poisson process, the times between successive events are independent and identical, but they can take any positive distribution. The Poisson process is thus a particular case of a renewal process.
The mathematics of the distributions of number of events in a period (equivalent to the Poisson distribution for the Poisson process) and the time to wait to observe x events (equivalent to the Gamma distribution in the Poisson process) can be quite complicated, depending on the distribution of time between events. However, Monte Carlo simulation lets us bypass the mathematics to arrive at both of these distributions.