The new ASP.NET Web API is a continuation of the previous WCF Web API project (although some of the concepts have changed).
WCF was originally created to enable SOAP-based services. For simpler RESTful or RPCish services (think clients like jQuery) ASP.NET Web API should be good choice.
For us, WCF is used for SOAP and Web API for REST. I wish Web API supported SOAP too. We are not using advanced features of WCF. Here is comparison from MSDN:
ASP.net Web API is all about HTTP and REST based GET,POST,PUT,DELETE with well know ASP.net MVC style of programming and JSON returnable; web API is for all the light weight process and pure HTTP based components. For one to go ahead with WCF even for simple or simplest single web service it will bring all the extra baggage. For light weight simple service for ajax or dynamic calls always WebApi just solves the need. This neatly complements or helps in parallel to the ASP.net MVC.
Check out the podcast : Hanselminutes Podcast 264 - This is not your father's WCF - All about the WebAPI with Glenn Block by Scott Hanselman for more information.
In the scenarios listed below you should go for WCF:
WEB API is a framework for developing RESTful/HTTP services.
There are so many clients that do not understand SOAP like Browsers, HTML5, in those cases WEB APIs are a good choice.
HTTP services header specifies how to secure service, how to cache the information, type of the message body and HTTP body can specify any type of content like HTML not just XML as SOAP services.