Web Services - that's standard defined by W3C, so they can be accessed semi-automatically or automatically (WSDL / UDDI). The whole thing is based on XML, so anyone can call it. And every aspect of the service is very well defined. There's parameters description standard, parameter passing standard, response standard, discovery standard, etc. etc. You could probably write 2000 pages book that'd describe the standard. There are even some "additional" standards for doing "standard" things, like authentication.
Despite the fact that automatic invoking and discovery is barely working because clients are rather poor, and you have no real guarantee that any service can be called from any client.
Web API is typically done as HTTP/REST, nothing is defined, output can be for eg. JSON/XML, input can be XML/JSON/or plain data. There are no standards for anything => no automatic calling and discovery. You can provide some description in text file or PDF, you can return the data in Windows-1250 instead of unicode, etc. For describing the standard it'd be 2 pages brochure with some simple info and you'll define everything else.
Web is switching towards Web API / REST. Web Services are really no better than Web API. Very complicated to develop and they eat much more resources (bandwidth and RAM)... and because of all data conversions (REQUEST->XML->DATA->RESPONSE->XML->VALIDATION->CONVERSION->DATA) are very slow.
Eg. In WebAPI you can pack the data, send it compressed and un-compress+un-pack on the client. In SOAP you could only compress HTML request.