What is the real difference between an API and an microservice?

The majority of the answers is based on the old-school understanding of API as a programmatic interface. Nowadays, this meaning is melted and start confusing people becuase some developers started (for simplicit or by mistake) interpred the API of an application as the application per se. In such case, it is impossible to distinguish between the modern API and Microservices. Nonetheless, we can say that an API-application can comprise many Microservices, the most of which interact within the application via Microservice's APIs while others may expose their APIs as Applications's APIs. Also, a Microservice (as a service) may not include other Microservices (services), but may orchestrate a composition of Microservices via API-bases invocations. Applications may contain Microservices but, in the best practices, may not contain other Applications.

In layman's term, if you have a web API server and you split them into several independent mini servers, use a proxy-server and load-balancer to clusterize them, and (optionally, give each a separate database entity), that is a microservice architecture.