La pile and la batterie are both translations for a battery. Is the distinction that la batterie can be charged and is built-in to devices like a cell phone, and la pile is the round object that's used, for example, in a flashlight. It can not be charged, and needs to be replaced when it dies.
Freeform Writing Exercise B2
In France, the word 'pile' is used to describe the small items you buy usually in multiples that you can recharge or recycle.
In a car, you will have 'une batterie' and surprisingly the same for a computer but the small round battery in a computer would be 'une pile'.
Take a look at the following page which I found particularly informative -
Hope this helps!
Both can mean battery (I'm not sure if there's a distinction between rechargable ones and ordinary ones). But la batterie is also drums and an array of stuff (a battery of games = a ton of games). La pile is also a heap or pile.
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