Any headphone that uses a headphone jack is analog in nature, and needs a separate digital to analog converter and an amp. That's what you bought as an external amp and DAC, and it's also what's built into your computer past the headphone jack. They're functionally both the same thing. If you were using just a vinyl player for example, you don't need a DAC at all as there's no digital signal to convert to analog. You'd just need the amplifier.

A headphone that uses USB as the interface has an internal amp and DAC. It's doing the work of your external amp and DAC, just inside the headphone, which allows you to plug it in with USB. It's still having to convert the digital signal to an analog one and amplify it, but it happens inside the headphone.