les baskets -- sneakers
Les chaussures de sport -- sport shoes

Les baskets is used the same way you would use "sneakers" in English. Les chaussures de sport is a more general way to mean footwear for sport.

Les baskets are specific to Basket Ball at least that was the original design intention.

Les chaussures de sport are sneaker/trainers more general sports-related, not specific.

I suspect these days there is more of a fashion element at play rather than outright specific to sport usage.