Consistency is usually a desired property for an estimator. We have the definition of consistency for an estimator $T_n$ for $\theta$, stating that it converges in probability to $\theta$, and the definition strong consistency, which asserts that it converges $\mathbb{P}_{\theta}$-almost surely to $\theta$. The latter implies the first.
I have no problem with the mathematical definitions or how to work with them. My question is distinguishing practically these two types of consistency.
For instance, suppose I have $T_n$, which is strongly consistent to $\theta$, and $R_n$, which is consistent, but not strongly consistent. What do I gain, in practice, from the fact that $T_n$ is strongly consistent?
If I made a simulation study to observe the convergence of both estimators, would there be a noticeable difference? Can $R_n$ have a better convergence rate than $T_n$?
