Simply put, the difference between a protocol and an API are that a protocol is how you communicate, while an API is a way for software to not just communicate with a server but get it to do things as if the software were a human using different, less efficient means.

Let's take pre-Elon Twitter for example: Twitter's API allowed automated programs to tweet things without simulating a browser, "clicking" on "new tweet", putting text in the text box and then "clicking" on "send". Instead they could directly send the text and identification info to a part of Twitter's API. This communication with the API happened through a protocol, most likely HTTP/HTTPS, because protocols have in-built ways to handle communication failures, encryption, etc.