You should check the MySQLdatabase is set to the recommended character encoding as Thorsten suggests, a warning is shown in the Environment report if this isn't configured.
Generally backing up and restoring between Moodles with different databases types should work except there is a technical restriction specific to restoring to MySQL (and MariaDB): MDL-9391. I've seen this problem with quizzes where PostgreSQL allowed answers with units with different cases which couldn't be restored to MySQL as its UNIQUE constraint treats e.g. "cm" and "CM" as identical.
Walter: I'd suggest enabling debugging (Debug messages: DEVELOPER, Display debug messages: Yes) on the site you're restoring to, which should show more information when the database error occurs.