Conceptual difference between Poisson and uniform distribution

I feel very stupid asking this question, because they're obviously different concepts, but I can't understand why. Every textbook I read has them both thoroughly explained, but at some point I can't grasp WHY they're different. Where do they defer?

Basically, what I can see is that in any Poisson process, if $N(t)$ is the amount of successes in a interval $[0,t)$, then $N(t_i)$ is independent of $N(t_j)$, $t_i$ and $t_j$ being sections of the interval, having the same length.

Besides, successes are distributed uniformly. The probability of a success does not depend on its position in the interval, but only its size.

I don't understand why those characteristics are not describing a Uniform distribution too. I mean, where is the line drawn?

Thank you very much for reading, and I'm sorry if you didn't understand something i said: English is not my prime language, and I acknowledge I have some problems using it.

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.