MPEG audio sound quality is highly dependent on two things: the bitrate, and the encoder. The particular compression layer (I, II, or III) in use will primarily only affect the lowest bitrate you can achieve while maintaining comparable audio quality.

All three layers make use of a 32-band synthesis filter when decoding to produce actual audio samples. The three layers differ in how the subband samples are computed from the bitstream.

Layer I is the simplest; each subband sample is read directly from the bitstream. In Layer II, a more efficient coding scheme is used to represent the samples, but not all subbands can be represented: a limit on the highest possible subband containing samples is imposed depending on the sampling frequency and bitrate. For example, at 44.1kHz, 64kbps/channel (128kbps stereo), Layer II can only encode subband samples for subbands 0-26. The highest subbands (27-31) can never have bits allocated to them at this frequency and bitrate.

Layer III uses the most complex coding scheme and includes additional tricks to maximize use of the bitstream, but the output is the same as it is for Layers I and II: samples for each of the 32 subbands.

All of your audio quality, then, is not necessarily dependent on the particular compression layer in use, but rather on the bitrate and on the quality of the encoder's psychoacoustic model, which determines how that bitrate is utilized. A possible exception here is Layer II because it can never reproduce frequencies corresponding to subbands 30 or 31 at any sampling frequency or bitrate. (I doubt most people would be able to notice, though.)