There are three basic ways to integrate SMF into your existing website and/or other scripts.

1. SSI.php can be used if you want to use SMF functions, variables, security, etc on external pages. This includes logins, access, revent posts, etc. This is not an "integration with an external system" so much as displaying and using SMF on pages outside the forum.
2. Integration hooks can be used if you want SMF to be the primary system, but you need to pass data to another script in the middle of an SMF process (i.e. new user signing up in SMF registration -> pass the user data to another script, run some more processes on that script and then return to SMF's routines)
3. The API can be used if you want the opposite of the hooks. The external script is the primary system and the API allows you to pass the data collected by that system into SMF's functions and database.