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.

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 overlap. A 'Sneaker' is a sporty, comfortable shoe that people often wear in everyday life. 'Sportschuh' can also mean this kind of everyday shoe, but the word can also refer to a shoe that is really made for sports. So 'Sportschuh' is somewhat broader. The words can be confused because many sneakers look sporty. If the shoe is part of an outfit, 'Sneaker' often sounds more natural. If the context is training, a gym, or a specific sport, 'Sportschuh' fits better.

Im Fitnessstudio darf man nur mit sauberen Sportschuhen trainieren.
At the gym, you may only train with clean sports shoes.
Diese Sneaker passen gut zu meinem neuen Sommerkleid.
These sneakers go well with my new summer dress.