1) Web API vs REST API: Protocol
Web API supports protocol for HTTP/s protocol and URL requests/responses headers that enable services to reach various clients through the web. On the other hand, all communication in the REST API is supported only through HTTP protocol.
2) Web API vs REST API: Formats
Although APIs perform identical tasks, a Web API provides flexibility to any style of communication. Whereas a REST API can take advantage of using REST, SOAP, and XML-RPC for communication.
3) Web API vs REST API: Design
As Web APIs are lightweight architecture, they are designed for gadgets constrained to devices like smartphones. In contrast, REST APIs send and receive data over systems making it a complex architecture.
4) Web API vs REST API: Support
Web API can be hosted only on an Internet Information Service (IIS) or self that supports XML and JSON requests. In contrast, REST API can be hosted only on IIS that supports standardized XML requests.