Probably mostly the latter. The headphone adapter with recent iPhones measures remarkably well for such a tiny device: https://www.head-fi.org/threads/what-is-the-sound-quality-of... (scroll down a little for lots of geeky measurements).

It's hard for our brain to comprehend that such a tiny gadget can sound so good, so I suspect that accounts for a lot of the perceived difference. Also, some music can sound better with a poorer output device by e.g. smearing noise in the treble or rolling off excessive bass.

I also have a 6th generation iPod Nano, an iPhone 5 and an iPhone X. I can hear the difference between the Nano and the iPhone 5.

The difference is not much. iPhone has slightly better highs and slightly tighter lows. The headphones I'm using is Sennheiser MM70i (They use the same buds with CX400-II, I also have these).

I didn't found the difference with "critical listening". One night I just started listening it, and it sounded different to my iPhone. Then I played the same song on the iPhone and found out the difference.

The difference maybe due to different generations of DACs, different OPAMPS, different power budget, different decoders, etc. As I said before, the difference is not that much, but normal for the evolution of these types of electronics. Nano honestly sounds impressive for a device that compact.

P.S.: I used to play euphonium in a large band and double bass in a symphony orchestra, so I have trained my ears for separation and tracking of instruments and rhythm. This is why I'm a bit more sensitive to these things than most of the people around.

I've always wondered if the headphone adapter contains a DAC or if it just acts as a breakout box for a DAC already in the phone. I found some people claiming it does, but none that I can see cite a source for it.

As far as I know, lightning connectors (as opposed to their 30 pin predecessors) carry no analog signals, so the DAC necessarily has to reside in the headphone adapter.

Could you have a couple friends help you set up some double blind comparisons? I'd be interested if learning the results.

Forgot to mention in my other response... the volume is the biggest influence in perceived sound quality. Equalising that in such devices can be tricky in an impromptu setup.

Then maybe you don't need to equalize it? Run your experiment with both sources at a wide range of volumes. If there's actually a noticeable difference, it should come across in the data.