What is the difference between a "function" and a "procedure"?

A function returns a value and a procedure just executes commands.

The name function comes from math. It is used to calculate a value based on input.

A procedure is a set of commands which can be executed in order.

In most programming languages, even functions can have a set of commands. Hence the difference is only returning a value.

But if you like to keep a function clean, (just look at functional languages), you need to make sure a function does not have a side effect.

This depends on the context.

In Pascal-like languages, functions and procedures are distinct entities, differing in whether they do or don't return a value. They behave differently wrt. the language syntax (eg. procedure calls form statements; you cannot use a procedure call inside an expression vs. function calls don't form statements, you must use them in other statements). Therefore, Pascal-bred programmers differentiate between those.

In C-like languages, and many other contemporary languages, this distinction is gone; in statically typed languages, procedures are just functions with a funny return type. This is probably why they are used interchangeably.

In functional languages, there is typically no such thing as a procedure - everything is a function.

Example in C:

// function
int square( int n ) {
   return n * n;
}

// procedure
void display( int n ) {
   printf( "The value is %d", n );
}

Although you should note that the C Standard doesn't talk about procedures, only functions.

In general, a procedure is a sequence of instructions.
A function can be the same, but it usually returns a result.