Say that I have two uncertain parameters, A and B. I estimate each population's mean and variance using estimators of the mean, $Q_A:=\frac{1}{N_A}\sum_{i=1}^{N_A}a_i, a_i \sim A$ and $Q_B:=\frac{1}{N_B}\sum_{i=1}^{N_B}b_i, b_i \sim B$, and variance, $\hat{s}_A:=\sum_{i=1}^{N_A}\frac{(a_i-Q_A)^2}{N_A}$ and $\hat{s}_B:=\sum_{i=1}^{N_B}\frac{(b_i-Q_B)^2}{N_B}$.
Say that I want to estimate $\frac{{\mathrm{Var}}(A)}{\mathrm{Var}(B)}$. If I'm doing Monte Carlo simulation and $N_a$>$N_b$, then I can either approximate it using
$\frac{{\mathrm{Var}}(A)}{\mathrm{Var}(B)} \approx \frac{\hat{s}_A}{\hat{s}_B}$,
or I can approximate it using the identical samples,
$\frac{{\mathrm{Var}}(A)}{\mathrm{Var}(B)} \approx \sum_{i=1}^{N_B} \frac{(a_i-Q_A)^2}{(b_i-Q_B)^2}$.
Is there any advantages or disadvantages to one of these approaches to the other? Is one expected to be more accurate?
You can't tell, just from the information you've given us, which approach will yield a better estimate. It depends on the correlation between $A$ and $B$.
In the extreme case that $A$ and $B$ are the same quantity, the second approach will clearly be better, whereas if there's no correlation between $A$ and $B$ (so the event of $B$ deviating a certain amount from its mean for a certain sample is independent of $A$ deviating by a certain amount from its mean for the same sample), then the first approach would be better.