Although you can do most of the things a stored procedure can do with simple ad hoc Transact-SQL code, stored procedures have a number of advantages over ad hoc queries, including
Execution plan retention and reuse
Query autoparameterization
Encapsulation of business rules and policies
Application modularization
Sharing of application logic between applications
Access to database objects that is both secure and uniform
Consistent, safe data modification
Network bandwidth conservation
Support for automatic execution at system start-up
I'll touch on each of these as we go along.