Question

When should I use a function rather than a stored procedure in SQL, and vice versa? What is the purpose of each?

Answer 1

Functions are computed values and cannot perform permanent environmental changes to SQL Server (i.e., no INSERT or UPDATE statements allowed).

A function can be used inline in SQL statements if it returns a scalar value or can be joined upon if it returns a result set.

A point worth noting from comments, which summarize the answer. Thanks to @Sean K Anderson:

Functions follow the computer-science definition in that they MUST return a value and cannot alter the data they receive as parameters
(the arguments). Functions are not allowed to change anything, must
have at least one parameter, and they must return a value. Stored
procs do not have to have a parameter, can change database objects,
and do not have to return a value.