Shell thickness and bearing edge are big factors in sound. But there are great sounding kits with sharp 45 degree edges, and great ones with rounded 30 degree edges. I prefer rounded edges for a more vintage sound, but it's just preference.
Hoops and hardware play a role, lighter helps with resonance if that's what you want, but it's obviously easier to experiment with different hoops after the fact vs shell construction.
While great sound is subjective, I think quality hardware that allows for easy, precise tuning is something that separates lower level kits from better gear.
That pretty much sums it up.
Other factors are shell geometry that should match the style of music you play - I play jazz and blues primarily, so 14x20 (or x18) bass drums, 8x14 and 14x14 toms are the norm for me. However, on the subject of floor toms, square sizes (same depth and diameter like 14x14 or 16x16) are not as easy to dial in tuning for a room than, say, 12x14 or 14x16. Also, as redbeard77 stated, bearing edges are important, and especially so for the style of music. Sharp 45s lack the warmth of rounded 30s, but the rounded 30s lack the attack and projection that some music requires. Heads and hoops are super critical. Some music (and situations) require a lot of attenuation, whereas others require a lot of open resonance. Studio playing requires a lot more focus and control than live playing, and heads, hoops & tuning (and bearing edges) all factor in moreso than material.
Bottom line: let the main style of music you play dictate the elements of design.