What are the differences between "Stored Procedures" and "Stored Functions"?

Officially, PostgreSQL only has "functions". Trigger functions are sometimes referred to as "trigger procedures", but that usage has no distinct meaning. Internally, functions are sometimes referred to as procedures, such as in the system catalog pg_proc. That's a holdover from PostQUEL. Any features that some people (possibly with experience in different database systems) might associate with procedures, such as their relevance to preventing SQL injections or the use of output parameters, also apply to functions as they exist in PostgreSQL.

Now, when people in the PostgreSQL community talk about "stored procedures" or "real stored procedures", however, they often mean a hypothetical feature of a function-like object that can start and stop transactions in its body, something that current functions cannot do. The use of the term "stored procedure" in this context appears to be by analogy to other database products. See this mailing list thread for a vague idea.

In practice, however, this distinction of function versus procedure in terms of their transaction-controlling capabilities is not universally accepted, and certainly many programmers without database bias will take a Pascal-like interpretation of a procedure as a function without return value. (The SQL standard appears to take a middle ground, in that a procedure by default has a different transaction behavior than a function, but this can be adjusted per object.) So in any case, and especially when looking at questions on Stack Exchange with a very mixed audience, you should avoid assuming too much and use clearer terms or define the properties that you expect.

In terms of DDL, Postgres does not have procedure objects, only functions. Postgres functions can return value(s) or void so they take on the roles of both functions and procedures in other RDBMSs. The word 'procedure' in the create trigger refers to a function.

In terms of the Postgres documentation, 'procedure' is also a synonym for the database object called a function, eg: "A trigger procedure is created with the CREATE FUNCTION command".

Trigger 'procedures' do have particular rules: they must be declared as a function with no arguments and a return type of trigger. Example here.

The terms "stored procedure" and "stored function" are used interchangeably in PostgreSQL and are generally taken to mean the same thing. Other databases may differentiate between a procedure and function (much like how VB differentiates between subroutines and functions).

As long as a function in PostgreSQL returns something that resembles a table, you can use the output of that function as if it were a standard table. The CREATE TRIGGER syntax is a bit confusing, but I suspect it may have been in place before the ANSI standard was finalized. I only have a copy of SQL:2003, so I can't do much more than speculate why the nomenclature is weird.

TL;DR version: with PostgreSQL "procedure" is equivalent to "function".