Web API provides a REST-friendly HTTP based API. Web API uses the patterns of MVC and is going to be very familiar to ASP.NET MVC developers. Web API can leverage the capabilities of HTTP as an application layer protocol, returning resources in multiple representations (XML, JSON, HTML etc.) according the the client's request headers.
On the other hand WCF webHttpBinding uses the patterns of WCF, and is going to appeal more to the WCF developer - ServiceContracts, OperationContracts, comprehensive (or overweight, depending how you look at it, config file), ability to self-host outside of IIS.
One of the things I like about Web API is the ability to use dynamic types to escape the constraints of the type system. I also like the default exception behavior in Web API - contrast WCF webHttpBinding where, by default, exceptions bubble up as HTTP 500 + an HTML payload (yuk!).
Its nice to have the choice between two excellent technologies here. I wouldn't describe Web API as 'newer' or 'better' that WCF, as this implies its a replacement technology and that WCF webHttpBinding is legacy, which I don't believe is true.
I chose to use WCF webHttpBinding recently to expose a JSON API for an existing WCF SOAP service. I believe it was a good choice because it fitted that style of that existing solution and minimized the amount of change required.