Shouldn't it be the same thing? You either have the DAC on your phone convert the digital music file to an analog signal and send it over the aux cord to the speakers in the headphones, or have the digital file sent over Bluetooth and converted by a DAC in the headphones, right? It's not like you're plugging your headphones into a record player.

> have the digital file sent over Bluetooth and converted by a DAC in the headphones, right

This is not how Bluetooth wireless audio works. PCM audio is re-encoded on-device into any one of a few Bluetooth-capable codecs that is then streamed to the client device. This is a primary cause of latency.