Well I would assume FP is just a special case of Poisson process...

A Poisson process is a way of counting independent exponential arrivals.

For definiteness:

Suppose posters arrive at this thread in independent exponential times with rate per hour.

Set the time of the arrival.

Then the number of arrivals by time is a poisson process

There are many interesting properties about this counting process, but ultimatleyit is used to count the number of arrivals of independent exponentials.