The easiest way to make this clear to a learner is to focus on the objects. John and Peter played a game of chess. John was the winner. Peter was the loser. John won the game. John beat Peter.

"win" does not involve the "loser". You can only "beat" someone else. If I say I win a race, I am saying nothing about the others. I can beat another person, but maybe not "win" (because others beat me). I hope this helps.

I would personally draw some simple pictures to explain it. With win/won showing the person dreaming of a 1st place trophy or hitting the jackpot at something and 'beat' showing a picture of someone passing another person over the finish line with arrows between them....together with a little phrase of course.