What is the difference between beaten and completed?

The difference between the beaten and completed is simple.

Beaten means completing the main story of a game or some other main goal.

Completed however can mean many different things depending on the game:

For some games completed could mean simply getting every achievement on platforms such as Steam or PSN, 100%ing the game's achievements or getting the platinum trophy on Playstation.

For games that don't have achievements, this could mean something difference to you, maybe the game makes it easy and has a completion stat somewhere in the menu, maybe it's completing every side quest or exploring the entire map. It's up to you!

However, generally the difference between the two, is "beaten" means you completed the main story, and "completed" means you completed both the main story and every optional extra, from side quests or optional bosses/challenges.