I was asked to explain this phenomenon using the example of logistic model during the next class (online of course). I do not have problems with analyzing both types of logistic model and I can certainly say the behavior differs, especially since
$$p_{n+1}=p_n+rp_n(1-p_n)$$
produces chaotic behavior for $r=4$. However I still struggle with answering the question WHY. It always appeared to me as a question more of philosophical nature rather than mathematical. I have always done the math and take my results for granted if my analysis was rigorous, that meant my solution / final result is right, that's it. I did not see a need for deeper understanding, nevertheless now I am forced to think it over ;)). Any help will be appreciated.
If $L$ is the Lipschitz constant of the right side in $\dot y=f(y)$, then the Euler method gives somewhat correct results for step sizes up to $r=0.1/L$ and a somewhat qualitatively correct behavior up to $r=1/L$. For the given $f(y)=y(1-y)$ you get $f'(y)=1-2y$ and thus $L=1$ on the interval $[0,1]$. This means that the step sizes where the discretization is still at least marginally related to the original ODE are all in the "boring" region of the logistic map where it has one attracting (non-zero) fixed point and no cycles.
$$ p_{n+1}=p_n+rp_n(1-p_n)=p_n(1+r-p_n)\implies\frac{p_{n+1}}{1+r}=(1+r)\cdot\frac{p_n}{1+r}\left(1-\frac{p_n}{1+r}\right) $$
Conversely, in the region of the parameter $r$ where the logistic map is "interesting" with branching limit cycles and chaotic behavior it has no longer any relation to the discretization of the logistic ODE.