I use the V6 stage with my iPhone 5 as well, but I much prefer listening out of my Audinst (HUD-MX1). I'm not a believer in cables and amps making improvements to sound, but the sound is indeed noticeably different between the two (I would definitely be able to tell them apart in a blind test)
If I were to go all subjective and write like a classic audiophile, I would tell you that what I can hear with the Audinst is a wider, more 3D soundstage that's fuller and slightly more transparent. Also more detailed while being less fatiguing in the highs. But that all can be attributed to having less treble causing me to turn it up and hearing more bass and mids letting me hear more detail in those ranges.
A simple and slight impedance difference can change the frequency response; i.e. cause a touch more treble in the balance. This can be perceived as the source sounding more 'detailed', but 'thinner in its sound'; while the other source would be perceived as sounding 'fuller' and 'less fatiguing', with 'more detail in the mids'.
So, I'll stay on the safe side and tell you for sure the Audinst has less treble presence (remember, every single review of the audinst praises it as neutral, not rolled off in the highs). I can definitely turn it up louder than the iphone without getting fatigued, yet I still hear all the treble detail. Listening through the Audinst is, I guess, a little closer to listening to a high end speaker system (i've heard dozens at TAVES, a local audio show held in a hotel, chock full of multi-thousand dollar speaker systems, some ranging to $100,000 for just the pair of speakers).
Between the two, I don't like listening to my iPhone as much because I can't hear the mids and bass as much - I prefer listening to a darker sound by nature of my youth. I certainly don't miss any detail or 3D soundstage awesomeness on the iPhone though, and is perfectly enjoyable on its own.
P.S. I've found my mid-2013 (Haswell) Macbook Air to have boosted treble and bass in its headphone out. I like my mids, so I don't like it very much; again this could be just because of the output impedance though.