  Let $\{Z_{1,n} , n\geq 0\}$ and $\{Z_{2,n}, n\geq 0\}$ be two supercritical
branching processes in different random environments, with criticality
parameters $\mu_1$ and $\mu_2$ respectively. It is known that $\frac{1}{n} \ln
Z_{1,n} \rightarrow \mu_1$ and $\frac{1}{m} \ln Z_{2,m} \rightarrow \mu_2$ in
probability as $m, n \rightarrow \infty.$ In this paper, we are interested in
the comparison on the two criticality parameters. To this end, we prove a
non-uniform Berry-Esseen's bound and Cram\'{e}r's moderate deviations for
$\frac{1}{n} \ln Z_{1,n} - \frac{1}{m} \ln Z_{2,m}$ as $m, n \rightarrow
\infty.$ An application is also given for constructing confidence interval for
$\mu_1-\mu_2$.
