 If $a_n$ involves factorials, then it's probably easier to compute $\displaystyle\frac{a_{n+1}}{a_n}$ than $\lvert a_n\rvert^{1/n}$, so the ratio test is probably best.
 If $a_n$ involves $n^\text{th}$ powers, then computing $\lvert a_n\rvert^{1/n}$ is easy, and the root test is probably best.
 The two tests give almost exactly the same information. If $R<1$, then $\rho <1$, if $R=1$ then $\rho=1$, and if $R>1$ then $\rho>1$. There are a few cases where the limit $\rho$ exists but $R$ doesn't, but these are rare.
