Differences (or similarity) between Radon-Nikodym and the Lebesgue-Stieltjes integral?

Question

What's the difference, or similarity, between the Radon-Nikodym theorem and the Lebesgue-Stieltjes integral? In Radon-Nikodym, under the proper assumptions, we can shove a function from the integral into the measure $$ \int_A f \frac{d\nu}{d\mu}d\mu = \int_Af d\nu $$ However, this also seems possible with a Lebesgue-Stieltjes integral since it seems like we can have $$ \int_A f g^\prime d\mu = \int_A f dg $$ As such, what's the fundamental difference between the two tools?