I read lots of about Web Api. For example i understand Web Service is a kind of Web Api or Web Api is more flexible.
But i didn't get that: Is Web Api future of Web Service?
For example one of our client needs data from our main database. Normally i use a Web Service for this -simple- purpose but this time i created a Web Api project. I got and service data plus i figured out how it works with Entity or Identity etc. But it's not simple as a web service. I think our client will think same thing also because of identity thing. So why should i prefer Web Api vs Web Service or should i prefer Web Api in this -simple- case?
This kind of depends what you mean by 'web service', but for now I'm going to assume you mean the old .net SOAP services.
If you are building something new today (September 2015) you are almost certainly better off using an asp.net web API. This is a standard REST-style service which can be called by almost any HTTP enabled client with no requirements for local software or understanding of how the service works, this is the whole point of the REST architectural style. I blogged a little about web API and REST here: http://blogs.msdn.com/b/martinkearn/archive/2015/01/05/introduction-to-rest-and-net-web-api.aspx
In your case of a simple service that adds CRUD operations to a database using entity framework. This can be very easily achieved with Web API. You can actually scaffold this whole thing based on a simple model.
To answer your specific question, Yes I believe that in eth asp.net world at least, web API is the future of web services. In fact web services have now been dropped in favour of web API.
Web API supports the .net identity model (I blogged on this here: http://blogs.msdn.com/b/martinkearn/archive/2015/03/25/securing-and-working-securely-with-web-api.aspx) and entity framework.
Hope this helps, if it does please mark as an answer or let me know of any more details you need.