The paid API is the premium version, the other one the basic version. The differences are described at LanguageTool - Premium.
In the documentation, it says that “maximum number of requests per minute: 20 (free) / 80 (Premium)”
I don’t understand how 80 requests per minute gives me the premium version and at the same time 250 calls per day. What kind of requests are these? How do they differ?
That plan of the API gives you 250 API calls per day to the premium version. The “per minute” restriction is an additional technical restriction to prevent our system from being overloaded.