Can anyone give example use cases of when you would benefit from using Redis and MongoDB in conjunction with each other?
Redis and MongoDB can be used together with good results. A company well-known for running MongoDB and Redis (along with MySQL and Sphinx) is Craiglist. See this presentation from Jeremy Zawodny.
MongoDB is interesting for persistent, document oriented, data indexed in various ways. Redis is more interesting for volatile data, or latency sensitive semi-persistent data.
Here are a few examples of concrete usage of Redis on top of MongoDB.
Pre-2.2 MongoDB does not have yet an expiration mechanism. Capped collections cannot really be used to implement a real TTL. Redis has a TTL-based expiration mechanism, making it convenient to store volatile data. For instance, user sessions are commonly stored in Redis, while user data will be stored and indexed in MongoDB. Note that MongoDB 2.2 has introduced a low accuracy expiration mechanism at the collection level (to be used for purging data for instance).
Redis provides a convenient set datatype and its associated operations (union, intersection, difference on multiple sets, etc ...). It is quite easy to implement a basic faceted search or tagging engine on top of this feature, which is an interesting addition to MongoDB more traditional indexing capabilities.
Redis supports efficient blocking pop operations on lists. This can be used to implement an ad-hoc distributed queuing system. It is more flexible than MongoDB tailable cursors IMO, since a backend application can listen to several queues with a timeout, transfer items to another queue atomically, etc ... If the application requires some queuing, it makes sense to store the queue in Redis, and keep the persistent functional data in MongoDB.
Redis also offers a pub/sub mechanism. In a distributed application, an event propagation system may be useful. This is again an excellent use case for Redis, while the persistent data are kept in MongoDB.
Because it is much easier to design a data model with MongoDB than with Redis (Redis is more low-level), it is interesting to benefit from the flexibility of MongoDB for main persistent data, and from the extra features provided by Redis (low latency, item expiration, queues, pub/sub, atomic blocks, etc ...). It is indeed a good combination.
Please note you should never run a Redis and MongoDB server on the same machine. MongoDB memory is designed to be swapped out, Redis is not. If MongoDB triggers some swapping activity, the performance of Redis will be catastrophic. They should be isolated on different nodes.