Looking at Microsoft's Managed Extensibility Framework (MEF) and various IoC containers (such as Unity), I am failing to see when to use one type of solution over the other. More specifically, it seems like MEF handles most IoC type patterns and that an IoC container like Unity would not be as necessary.
Ideally, I would like to see a good use case where an IoC container would be used instead of, or in addition to, MEF.
Best Answer
When boiled down, the main difference is that IoC containers are generally most useful with static dependencies (known at compile-time), and MEF is generally most useful with dynamic dependencies (known only at run-time).
As such, they are both composition engines, but the emphasis is very different for each pattern. Design decisions thus vary wildly, as MEF is optimized around discovery of unknown parts, rather than registrations of known parts.
Think about it this way: if you are developing your entire application, an IoC container is probably best. If you are writing for extensibility, such that 3rd-party developers will be extending your system, MEF is probably best.
Also, the article in @Pavel Nikolov's answer provides some great direction (it is written by Glenn Block, MEF's program manager).