Too many to have any control over -- you'll never spot the difference, at least not without destructive testing, microscopes, chemical tests, and extreme in-depth knowledge of industry formulations and techniques.
Electrolyte: solvent, salts (or acids or other ionic compounds), additives, impurities
Separator: thickness, material (cellulose paper or other?), possible reactions with electrolyte or electrodes
Electrodes and leads: thickness, grain structure, impurities, etching
Seal: thickness, material, impurities, mechanical fit, susceptibility and diffusion of solvent through material
Leads, assembly: base metal, plating, solder, flux
Probably even more things than these, and these are just high level outlines, not specific items and how to test them.
Probably a lot of things aren't very important, or never are (maybe sealing materials are well enough understood, and standardized, that they tend not to have problems by themselves?), and can be dismissed with fortunately little evidence necessary.
But other things are huge, and impossible to tell for any mere mortal. Supposedly the mid 2000s capacitor plague -- that affected many electronics manufacturers, big and small alike -- was a consequence of poor electrolyte formulation. I don't know the details, and a lot of that will be trade secrets anyway.
So, quantitatively speaking, it really is hard to beat simply having an epidemiological history of some product. You'll have no idea about the health of an individual -- has it been used for years of total operation, or did it spend most of its life in a dry, cool storage area? -- but you can at least anticipate likely problems from frequent offenders (like the mid-age classic Teks).
Tim