WCF Web API handles XML and JSON out of the box whereas you'll have to create your XML and JSON "by hand" (read: using the Serializers) when using MVC - this is only one of the benefits of WCF Web API over MVC.
Another one is the level of separation WCF Web API offers e.g. implementing your service logic vs. content negotiation.
Yet you can easily use IoC containers and unit test your APIs being created using WCF Web API.
WCF Web API mainly has been created to build ReSTful API's whereas MVC just allows it to create them too - thus with WCF Web API you'll feel more comfortable creating ReST APIs.
If you're planning to start a project from scratch as an Web (ReST) Api, you should start with WCF Web API.
If you're planning to start with a Website that also offers ReSTful Services, you should suggest MVC.
You should also regard this tweet from Glenn Block.