I own & play several wood (ply) & metal snares. Wood provides a warmth (less high frequency overtones, more mid-to-low frequency overtones) that, IMO, suits club-style jazz. Wood snare drums are also not as loud. However, wood snare drums don't "cut" through the mix of the other instruments like a metal drum does.
Of the wood ply snares, the shells range from 4mm thick to over 10mm thick. The thicker shells are louder and have a sharp, crisp
pop compared to a thinner shell.
Regarding type of wood, the softer the wood, the warmer and less cutting its sound is. Poplar was widely used in the 1950s and 60s. Then came birch, then maple, then bubinga; each increasing in hardness, and thus in brightness & sharpness of sonics.
As for rims, die-cast reduce overtones slightly and thus provide a more focused sound. Triple flange rims let the drum ring. I've not tried wood hoops, but they increase the diameter of the drum, so make sure your snare stand can hold it.
The throw-off and butt should be easy to operate and allow for plenty of slack to get the amount of buzz you want, and to allow for easy tuning. I really like the Tama "linear drive" strainer & butt (MLS50A/MLS50B), but it's not as quick of a release as the flip style found on other makes/models.
The diameter affects the pitch. A 14" will have a lower fundamental note than a 12". This lower tone won't sound as "thin" in the mix.
I have found that the shell depth affects volume/loudness more than tone. A shallow drum (2"–3") will sound more like
splat than a musical note. In the 5"–6" range the drum will have a good balance of body, loudness and cut. A deep snare (8"+) won't give you a "tom-plus-snare" sound unless you tune it very low & loose. Even then, it's not like a tom with snare wires, and with a loose resonant head the wires buzz sympathetically from any other drum or instrument (e.g., bass).
If you wonder about the number of lugs (8 vs. 10)
this excellent video will answer your questions (all their videos on snares are very informative).
I've bought used drums & cymbals from reverb.com and have had great luck, but my purchases have been from drummers or drum shops. They typically know more about the condition and value of what they sell.
Regarding older Slingerland and Ludwig drums, if they are in good-to-great condition they will cost as much or more than a new snare. This is a testament to the desirability of their sound.
If you're looking to produce a sound that was made in the 1960s (e.g., Joe Morello) then you may want a snare that is similar in design to what was used back in the day. If you're looking to produce a sound that is unique to you and your band, then you have many more options to choose from. Good luck!
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