The SQ isn't really less, in fact on a blind and volume-matched test between iPhone 4 and Note 2 a friend and I both slightly preferred the Note 2 overall (much to our surprise when we were done and could check our blind listening notes against the actual playback devices, we both heavily expected the iPhone to romp home). The difference in SQ you are appreciating will largely be a consequence of the volume output being noticeably lower on the Note 2 - in order to volume match them we had the iPhone down three steps from maximum volume.
Apps like PowerAMP will let you increase volume a little bit, but not a great deal, it may be enough for you though. Alternative solutions would be using a DAC/amp as Retrias mentions (what I do), or Rooting your Note 2 and installing one of the kernels with audio tweaks for increasing volume.