Conceptual difference between Poisson and uniform distribution

Not a stupid question at all!

For a Poisson process, if one and only one event occurs in the interval between 0 and $t$, then the timing of when the event occurs is uniform between 0 and t. The total number of occurrences $N(t)$ is a Poisson random variable. It's when we "zoom in" and look at a single occurrence that we observe a uniform distribution (or an exponential distribution if we're interested in the waiting time instead of the time of the occurrence).

The existing answer is good, but I will attempt a more precise one. This tutorial explains how a homogeneous Poisson process is similar to a set of $N$ draws from a uniform distribution and how they are different.

http://www.maths.qmul.ac.uk/~ig/MAS338/PP%20and%20uniform%20d-n.pdf

• They're different because a size-$N$ sample from a uniform distribution has exactly $N$ points, whereas a homogeneous Poisson process can produce any number of points. The total number of points is (discretely) Poisson-distributed.
• They're similar because if you sample the total number of points from a (discrete) Poisson distribution, and then sample their locations (aka arrival times) from $N$ iid uniform RV's, then the resulting point process is equivalent to a Poisson process sampled through any other method.