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Apple Watch almost certainly doesn't have wifi. We know this because we are told in Apple's video that if we go cycling we can use gps and wifi on the iphone to measure the distance travelled. This implies that the Watch doesn't have wifi.

The Watch certainly stores some music. The Watch likely has bluetooth, which is much lower energy than wifi. It is standard techology to stream music wirelessly over bluetooth.

So rather than going running with a giant phone strapped to your arm and a wire to your headphones, you can just run with the watch and have a wireless headphone.
 
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