Q-values are a great way to the make actions explicit so  you can deal with problems where the transition function is not available (model-free). However, when your action-space is large, things are not so nice and Q-values are not so convenient. Think of a huge number of actions or even continuous action-spaces.

From a sampling perspective, the dimensionality of $Q(s, a)$ is higher than $V(s)$ so it might get harder to get enough $(s, a)$ samples in comparison with $(s)$. If you have access to the transition function sometimes $V$ is good. 

There are also other uses where both are combined. For instance, the advantage function where $A(s, a) = Q(s, a) - V(s)$. If you are interested, you can find a recent example using advantage functions here:

> [Dueling Network Architectures for Deep Reinforcement Learning][1]

by Ziyu Wang, Tom Schaul, Matteo Hessel, Hado van Hasselt, Marc Lanctot and Nando de Freitas.

  [1]: http://arxiv.org/abs/1511.06581