As already written in the comments, they work faster and allow you to organize a rather cunning logic of working on data with your scripts. There is no ubiquitous practice (sometimes justified) when DBMS engineers describe all database operations in the form of stored procedures and provide their backend to developers as an API. Accordingly, the backend does not even fit into the database, does not answer for the correctness of each individual request, and generally focuses on other tasks, being engaged in the business logic of the application and almost without thinking about the data storage layer. So the developer knows that in order to obtain data on sales for the month, he pulls procedure X , and to initiate the closure of the financial period, he pulls procedure Y.
At the same time, stored procedures (in particular, MySql ) are very difficult to debug, plus they still contain part of business logic, i.e. what happens in your program becomes far from obvious, based on the application code. The very hell begins if a part of the same aspect of business logic is displayed both in the code and in the stored procedure, which can sometimes be observed in projects with a large legacy code. When some developer wrote stored procedures, then left the command, none of the other team members could write them, and later they continued to write the project, not using procedures, putting their logic into the applications, but not doing the normal code. Time passes, a new team is recruited for the project, and it needs to duplicate changes in business logic in several different places of the application plus in the stored procedure.
At present, backend developers, as a rule, are seated with various ORMs and this often ends their interaction with the base. If you have a separate person on the project who is ready to write and maintain stored procedures, providing you with only the described interface - with proper organization this can be beneficial for the development. Trying to do nothing to start using stored procedures in a combat project that previously did without them, I would not advise.