Script problems and a fractious relationship with Universal are said to be the cause of the split.
Two scripts have been penned for the film, and reportedly no one is fond of George Nolfi's take. Josh Zetumer came aboard to write a new one, a decision that irked Greengrass as he wasn't consulted as to the choice of writer. But that's a relatively minor point, as the real problem seems to be Greengrass' free-wheeling style, which has cost Universal millions and millions on every film he's made. While Bourne turned a fine profit, the studio is concerned about Green Zone, which is now estimated to have cost $150 million due to all of Greengrass' famous tinkering. This is a film that doesn't carry the Bourne brand name, and deals with the very un-blockbuster topic of Iraq.
So, Greengrass and Universal have been butting heads, as he wants to continue his usual approach on Bourne 4, and they want him to keep the budget under control. Thus, he's upset at the way he's been treated, and has now walked away. Rumors swirl that he might return, but there's just as much feeling that the studio could hire a new director. This leaves everyone uncertain as to whether Damon would return since he's been famously loyal to Greengrass. For now, everyone is watching the trades to see what'll be announced (and with Damon doing Invictus press, maybe it'll come up unofficially), and hoping for the best. The last thing anyone wants is Bourne without the guys that made it great.
UPDATED: Head after the jump for a statement from Paul Greengrass
Greengrass on the Bourne decision: "You won't find a more devoted supporter of the Bourne franchise than me. I will always be grateful to have been the caretaker to Jason Bourne over the course of The Bourne Supremacy and The Bourne Ultimatum. I'm very proud of those films and feel they express everything I most passionately believe about the possibility of making quality movies in the mainstream. My decision to not return a third time as director is simply about feeling the call for a different challenge. There's been no disagreement with Universal Pictures. The opportunity to work with the Bourne family again is a difficult thing to pass up, but we have discussed this together and they have been incredibly understanding and supportive. I've been lucky enough to have made four films for Universal, and our relationship continues. Jason Bourne existed before me and will continue, and I hope to remain involved in some capacity as the series moves on."