One obvious tech change would be going from analog to digital input, but makes less difference than tech that was already around at the time which won't be present on an inexpensive 4.0+ ch. setup.
Better designed cabinet using more expensive materials, better sound dampening, better drivers, more powerful amp, are all things contributing to sound quality that existed 20 years ago but then as now, they cost more money.
Today it is often "let's make a cool looking enclosure then put some speakers in it", rather than let's make an acoustically good enclosure designed around good drivers. The former is more about molding plastic. The latter takes experienced engineering.
If you want tonally balanced speakers rather than boomy bass, make sure the amp has sufficient output power (limits distortion) and the speakers are more of a studio monitor than something designed for gaming. After that, increasing the woofer size allows for deeper base that can remain more controlled, accurate.
A sound card may result in better analog output but will do nothing for digital output over motherboard integrated, except for any environmental gaming enhancements. However the analog improvement is often heard more from directly driving headphones, and along the same issue, would be better if a headamp was inserted between the sound card and the load (the headphones).
In the end it's about how much you want to spend. If you want at least 4 channels and $100 or less, maybe even $150 or less, you may get the most bang for the buck from a "computer" intended set.
What yerpuh linked looks like a good value in that price range, though sets like this with an amp in the sub, tend to have shorter amp lifespans. I wouldn't expect to get 20 years out of it, but that's such a long time that there are many class D sets out there today that I wouldn't expect to get 20 years out of purely because the capacitors won't last in a switching digital supply. Odds are your current set works still because it's class A/B and uses an unregulated big old 60Hz AC/DC transformer based PSU.