Typically, an increase in power (Watts) can improve dynamics, bass control and sometimes bass extension IF AND ONLY IF the headphone (or speakers) were not efficient enough for the lesser amp. In other words, if the lesser amp was already powerful enough to maximize these traits in a sufficiently efficient headphone, increasing the power will not improve the traits. It's when you've been under-powering a headphone that it can "scale" with an increase in power.
So, looking only at the huge difference in power between the iDSD and the PHA-3, ignoring all other features and specifications, we can easily conclude that there's a huge difference in the number of headphones that can be driven with sufficient authority to maximize their performance in terms of bass control and dynamics. If the amp is weak enough relative to the inefficiency of the headphone, even the mids can get muddy or woolly sounding.
Balanced output can eliminate cross-channel impedance fluctuations that can occur on single-ended cables that share a common ground - which can allow the two drivers to influence each other subtly, but this influence varies with the headphone design, where an almost purely resistive planar magnetic might be less vulnerable than a dynamic headphone for example.
That's the theory, anyway, but in my admittedly limited experience, I can't say I have ever heard the impact of cross-channel impedance fluctuations on single-ended cables. I' don't even know for what I should listen, except perhaps for a loss transparency. I do know, however, that there are many expensive headphone amps which provide plenty of power for inefficient headphones at an unbalanced TRS jack, where neither the designers nor the owners of these amps find themselves wishing it had balanced output. A lot of trusted, experienced reviewers are perfectly content with plugging their flagship headphones into TRS jacks.
Mike