This is only true if events are independent. Suppose we're modeling rider arrivals at a bus stop. The bus comes once an hour. Does the exponential distribution adequately model how long you need to wait for another person to show up? If a Poisson process is an appropriate model, the expected value of the number of people arriving in the 20 minutes after the bus departs is the same as the expected value of the number of people arriving in the 20 minutes before. Clearly not. Rider arrival rates depend on how long it is until the bus is supposed to depart. Indeed, most arrival processes are not ideal Poisson processes, but it may be a better approximation in some cases than in others.
We use Poisson processes for two reasons:
1. They're often a good enough approximation of reality to make useful predictions.
2. They're mathematically tractable.
Unfortunately, our reasons for using Poisson processes are often more (2) than (1).
None of this before/after bus mess.
That's funny -- my field has the same thing with a different distribution. Analyzing stochastic processes is much easier if you assume an exponential distribution. It's one of my criteria for whether somebody's giving a bad job talk. If an unfounded assumption that waiting times are exponentially distributed shows up in the first three slides, the rest of the presentation is probably B.S. Even if the presenter used Beamer and filled the slides with beautiful equations. (The exponential distribution of waiting times is basically an assumption that events are independent.)
Edit: Especially if the presenter's slides are full of beautiful equations.