ODesk was the last of the “Big 3” outsourcing portals (ODesk, Elance, Freelancer.com) that did not offer escrow services and arbitration for fixed price projects. One might remember: A great arbitration service was one of the key features of RentaCoder, one of the famous grandfathers of all current “online staffing” portals. RentaCoder (later renamed to vWorker) was acquired by Freelancer.com in November 2012 for a price in the millions.

Back to Odesk - their announcement says:

All new and existing fixed-price contracts are now fully protected by escrow. Deposit funds into a secure escrow account at the start of each milestone. The funds will be released as payment to your freelancer once you confirm that the milestone is complete.  

With this announcements all three major freelance IT worker marketplaces offer now escrow. But any escrow service is only as good as the arbitration process attached to it.
When it comes to a dispute you want to have a clear process and knowledgeable arbitrator that decides the dispute. I would assume that since ODesk was acquired by Elance in December 2013, both services will the same agents in the background. But that is just a guess at this point.

Note: In May 2015 ODesk was rebranded to Upwork.