What is the difference between an IORef and an MVar?

MVar is, like you said, targeted at multithreading while IORef can be used both as mutable variable in a single threaded program or as a synchronization construct in a multithreaded program.

IORef can be used together with atomicModifyIORef to get compare-and-swap (CAS) behavior: writers and readers can synchronize on a single pure value, stored in the IORef. Readers use readIORef to read a value and writers use atomicModifyIORef to write a value. Note that atomicModifyIORef doesn't let writers perform any side effects inside the critical section (i.e. they can only use a pure function when atomically changing the value).

MVar allows you to implement arbitrary critical sections (using withMVar), that may contain side effects. They can also be used just like an IORef (as described in the previous paragraph), but at a higher cost.

If you want an intuition for what kind of semantic IORef implements its the same as the CAS semantics Rich Hickey describes in a talk on Clojure's concurrency model: http://www.infoq.com/presentations/Are-We-There-Yet-Rich-Hickey

Edit: In addition, you cannot run into deadlocks using IORef (but there still might be contention, causing retries).