
der Sportschuh
sports shoe
A shoe that has been specially designed and made for athletic activities.
Für das Fitnessstudio brauche ich neue Sportschuhe.
I need new sports shoes for the gym.
sports shoe / sneaker
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sports shoe
A shoe that has been specially designed and made for athletic activities.
Für das Fitnessstudio brauche ich neue Sportschuhe.
I need new sports shoes for the gym.

sneaker
A comfortable shoe in a sporty style that is worn in everyday life as a fashion accessory; a sneaker.
Sie trägt zum Anzug bewusst lässige Sportschuhe.
She deliberately wears casual sneakers with her suit.
The word is a simple compound made up of 'Sport' (sport), the linking -s, and 'Schuh' (shoe). So it literally describes a shoe made for sports.
Ich habe mir für das Training neue Sportschuhe gekauft.
I bought myself new sports shoes for training.
The context shows whether the shoe is meant for sports (meaning 1) or as a fashion accessory (meaning 2). If it is about activities like 'laufen' (running) or 'trainieren' (training), the first meaning is intended. If it is about 'Alltag' (everyday life), 'Büro' (office), or 'Mode' (fashion), the second is meant.
Diese Sportschuhe sind nicht zum Joggen geeignet, sie sind nur für die Freizeit.
These sneakers are not suitable for jogging; they are only for casual wear.
'Sportschuh' (sports shoe) is a general term. In everyday language, more precise words are often used that name the sport, e.g. 'Laufschuhe' (running shoes), 'Fußballschuhe' (soccer cleats), 'Wanderschuhe' (hiking boots), or 'Hallenschuhe' (indoor sports shoes).
Für den Sportunterricht in der Halle braucht mein Sohn neue Hallenschuhe.
My son needs new indoor sports shoes for PE class in the gym.
sneaker
Sneaker and Sportschuh can both refer to a comfortable shoe in a sporty style. Sneaker usually emphasizes everyday use. Sportschuh can also refer to this kind of casual shoe, but especially describes a shoe designed for a specific sport.

Für das Lauftraining zieht Mia geeignete Sportschuhe an.
Mia puts on suitable running shoes for her running training.
Für den Stadtbummel zieht Mia bequeme Sneaker an.
Mia puts on comfortable sneakers for a stroll around town.