the better the converters, the better representation of the analogue sound you can get.
this could depend on the timing clock of the converters.. the better the clock, the more hope you having an accurate recording as the phase of the recording will not shift around giving a false reading. the usual aspects such as bit depth and sampling rate would also apply. to minimize quantize error :: which occurs when your converter plays 'join-the-dots' to form a waveform. it effectively just joins from one block to another, and inaccurate jumps from block to block creates a distortion is isnt really desirable for recording.
so the better clock and better a-to-d algorithm would determine a better recording. along with other aspects im sure.