I seem to recall it was low noise and distortion, perfectly flat FR or nearly so, and unusually good phase response to rather near the limits of the sample rate. Other than that you'll have to ask those who continue to use them in studios.
These cost $16k, and they were sold
at cost. Some serious engineering inside I'm sure. Converter hybrids, high supply voltages, DSP (for HDCD encoding and decoding if nothing else), the whole nine yards. The front panel design hasn't aged half-badly either, IMHO.
Specifications for the later Model Two (which went up to 192k, the Model One was 44.1/88.2 only) are rather terse, but the
HDCD Signal Processing section provides a glimpse of the complexity inside:
Eight Motorola 56009 DSPs and one Pacific Microsonics PMD-100 HDCD decoder ASIC control A/D conversion, sampling rate conversion, word length conversion, digital gain adjustment, HDCD encoding and decoding and D/A conversion.
Eight 56k DSPs.

I don't think I've ever sighted more than two DSP chips in any consumer-grade equipment.
Also, the thing seems to be built like an absolute tank:
Weight
Processor Unit: 35 lbs. (16 kg.)
Power Supply: 20 lbs. (9 kg.)
Custom ATA Case: 41 lbs. (19 kg.)
Now, as much fun as it is to marvel at the SOTA of days past, I sure am glad that merely
good converters have gotten way more affordable and power-efficient over the years. These days you can get a 120 dB(A) hi-res DAC / headphone driver the size of a pen and operating on USB bus power (if consumer-level with 1 Vrms out max) for $99. (
IL-DSP, based on
this guy.) That's pretty close to
"the computer is the bump in the power cord" terrain and rather amazing in its own right.