I can't find differences between $P(1+r)^n$ and $P(2.71828)^{rn}$ They told me $P(1 + r)^n$ can be used to calculate money interest, example:

You invert $15,000.00, 20% interest annual, for 3 years:
$15,000(1 + 0.20)^3$ = 25,920

And that $P(2.71828)^{rn}$ can be used to calculate population growth, example:

We have 500 bacterias, growing at the rate of 25% monthly, for 16 months:
$(500)(2.71828)^{(0.25)(16)}$ = roughly 26,968

I can't tell difference on when to use one over another. I mean different than seeing the keywords money and population. How do I tell when to use one over the another?
As far as I can tell, they both are something like:


*

*We have a starting value, a percent, a period of time


The difference is whether you try to model each discrete step or whether you choose a continuous model.
If you regard money, then you know that your interest will be calculated annually, so the discrete model will be exact.
If you have bacteria or population, you do not have any control about the number of babies or bacteria born at a particular time. You just know that you have enough people or bacteria, so that a statistical approach to the growth rate applies.
This means that you say that there are so many bacteria or people that you model it in a continuous way.
But you have to take care. In the continuous model, $r=0.25$ in your formula does not mean that you have a quarter more bacteria at the end of the month. This is a different $r$ from the one in the discrete model.
