I can't think of many at all, except that a WCF service is more complex, and thus risky, and you have to manage your own hosting, where IIS is designed to host, and has years more in the field than WCF.
My client wanted to use WCF as a web service for his users, people on laptops, to perform basic auth, and then CRUD operations regarding appointments for a given day. I m much more in favour of a Web Api, with built in auth, basically built in everything you need except the CRUD code itself.
Other aspects of the spec convince me he is yet another BA with very little exposure to real life software development.
What I have tried:
Knowing a Web API would be far simpler and safer, having implemented both for the same product before.