This is the way BBC explains the difference between "sport" and "games".
I hope it will solve your doubt.
sport is usually used in its singular form as a noun. my favorite sport is tennis, for example. when you use 'sports', it is like an adjective, for example 'sports facilities, sports equipment'.
a game is one event, sport is a free time activity that has dfiferent types.
What makes it a sport? In my mind, if it's in the Olympics it's a sport. However, that doesn't mean it will only ever be referred to as a sport. In the example above you see that game is used to refer to soccer, even though soccer is most definitely sport.
Game has a wider definition and applies to more past-time oriented/light hearted activites like board games, video games, card games etc. Often less physically active though that is not always the case, games involve a wider variety of mediums.
A sport is a physical activity carried out under an agreed set of rules, with a recreational purpose: for competition or self-enjoyment or a combination of these.
A game is a recreational activity involving one or more players, defined by a goal that the players try to reach, and some set of rules to play it.