What is the benefit of using stored procedures instead of SQL queries from an external connection?
One benefit, in terms of execution, is stored procedures store their query plan whereas dynamic sql query plans will not be stored and must be calculated each time the query is executed.
Is there any execution speed differences between them for small volume and big volume outputs?
Once the query plan is calculated, no, there is no speed difference.
Is there any benefits for the database management as well?
This is very subjective! In the past I worked at a place where ALL database access went through stored procedures so that they could lock down access to just the SPs. Other places I've worked didn't use stored procs at all because they are generally outside source control and problematic for developers who aren't SQL gurus. Also, business logic spread across multiple systems can become a real problem.