The difference has as much to do with the evolution of recording technology as it does with the gear.
One of the most overlooked facts of what I call the "analog era" is that all that old classic gear, the sounds the OP pines for now, was full of flaws, inconsistencies in component values, noise, and so forth. That applies to the instruments, amps, cabs, the recording gear, everything. All of those flaws and inconsistencies interacting, X bass through Y amp into Z board or whatever, all analog circuits with manual control, is what created the tonal characters of that era -- and it took a lot of effort to create them. I recall a story about Jimi Hendrix trying some of the first Muff pedals; he tried like a dozen before settling on a few he liked, because they all sounded different. Flaws and inconsistencies...
Now, in today's "digital era", processors and algorithms do much of the work, and the technology is much more consistent and precise. The drive from audiophiles has been to create the cleanest, most faithful, most precise sound reproductions possible, and that in turn has...well, I don't want to say "sucked the soul out" of music, but it has certainly affected it. Now it actually takes an effort to create a lively tone that stands out, to do something different, because the evolution of the technology has been toward making everything the same. Hence all us gearheads, and particularly us pedalheads, chasing the tones we hear in our heads, which (consciously or otherwise) are based on tones we remember from the past. Hence the constant debates over analog vs. digital pedals, passive vs. active basses, tubes vs. class D, and so forth. We seek the tones of the past, while the technology of the future actively fights us in doing so.
It is what it is, I guess...