For me an amp is about so much more than merely the volume level or amount of power, amongst other things it can have a major influence on the sound separation, soundstage and bass control with the volume being identical. I can drive the P9's from my laptop with a volume level which causes my head to explode, but it still sounds like ****. It's not just about getting the drivers to start vibrating, but more importantly to stop the vibration as the music demands. A mismatched amplifier will struggle most with this part, causing distortion and lack of soundstage. At the danger of repeating myself, it's not about the volume.
As an example, two years ago I got a serious pair of ($3k) full size speakers which sounded absolutely fantastic in the store. After having bought them and experiencing the first few days of confirmation bias induced euphoria, it quickly became apparent these sounded a lot worse than the ones in the store did. Muddy, unmusical, boring, distorted... Yet the (30 year old) amplifier it was paired with had recently been measured by an electrical engineer and was distortion free, while the power rating was identical to the amplifier used in the store. The volume was usually just below a quarter. After scratching my head for two months, the store I bought the speakers from was kind enough to lend me their amplifier for a few days. Immediately the sound improved to what I'd expected when I bought the speakers in the first place, so an amplifier got added to the system. Two years later I'm still incredibly happy with this combination, which long exceeds the confirmation bias period.
Now I firmly believe that buying a far more expensive amplifier for this set of speakers would only have resulted in a very minimal increase in sound quality. Once an amplifier is good enough for a certain set, then there is not much left to improve on. But having an unsuited amplifier can truly ruin the sound and make them perform way under their potential. I've experienced the same with the P9's - you could call the difference subtle if you want (a single instrument playing a low tempo tune sounds much the same for example), but as soon as the music becomes even slightly more complex the differences become huge - with amplifier it stays musical and you can keep all the instruments separated, without it becomes a mash of meaningless noise. One is fun and enjoyable, the other is hard work to listen to.
As you rightly point out, it could be the DAC. In all honesty, unlike with amplifiers, I don't have experience listening to differents DAC's side by side. But tom's hardware did quite some tests and found no meaningful difference in sound quality
between a 2$ and 2000$ DAC. So while I cannot exclude the DAC as the source of the sound difference, I do doubt it has so much influence.
Perhaps you're lucky and the iphone6 simply has a very good on board amplifier - I don't own one, I don't have access to one so I can't tell. All I can say is the sound from both my and laptop benefits immeasurably from using an external amplifier.