typically you'd use a database when there is a need to do so...i.e. there's more to the data than a single table with a single column, and you want to query/update the data in different ways.
as to performance...mysql is very fast...but for a single field append, you're not going to see any substantial difference in performance between mysql or plain file anyway.
one thing a database does give you, without lifting a finger, is locking