#### Many econometricians/statisticians view the "force of growth" as the most natural measure and therefore see the percentage change as the approximation

The reason for this usage is that logarithmic difference gets you the "force-of-growth" under continuously compounded growth, which is perhaps the most natural way to measure growth in a quantity over time.  (In different econometric contexts this ,ay have a more specific name such as the [force of interest](https://en.wikipedia.org/wiki/Compound_interest#Continuous_compounding), etc.)  


If you have a nominal growth rate $r$ and you compound the growth $n$ times per time-unit for $t$ time units then the total growth $p_n$ over the time period satisfies:

$$1+p_n = \bigg( 1 + \frac{r}{n} \bigg)^{nt}.$$

Continuous compounding occurs when we take $n \rightarrow \infty$ which then gives:

$$1+p_\infty = \exp(rt),$$

and $r$ now represents the "force of growth".  One of the nice things about measuring using the force of growth is that it is comparable for things growing over different time periods.  The total force of growth over $t$ time-units is $rt$, which is a linear function of the amount of time elapsed.  If you start with an amount $A_0$ at some time and then this grows to $A_t$ after $t$ time-units then you have $A_t = A_0 (1+p_\infty) = A_0 \exp(rt)$, which then gives:

$$rt = \log(A_t) - \log(A_0).$$

Your question amounts to asking why we would look at $rt$ instead of looking at $p_n$, with the former asserted to be an approximation of the latter.  However, it is actually somewhat more natural to see the force of growth as the important measure, such that the percentage change in $p_n$ is actually the approximation to the more important quantity.