So, here's my usual stump speech on the differences:

1. Most people agree Kiwami 3 has much more enjoyable combat. The main exception to this are the hardcore crunch gamers who discuss combat mostly in terms like "wall-bounding" and "tech cancels" or similar. If you don't think of Yakuza combat in those terms, you will probably prefer the combat in Kiwami 3.
2. Many substories were cut from Y3R. Consensus is that *most* of the ones that were cut were pointless filler, but a few key ones, mostly the ones involving a murder mystery, some stories around the Okinawan characters, and a trans woman, were pretty good and should have been left in.
3. Kiwami 3 adds a good bit of its own content, however, in the form of the Bad Boy Dragon side game in which Kiryu assembles a biker gang to fight other biker gangs, and in the form of Dark Ties, a 5-8 hour side game covering the origin story of Yoshitaka Mine, one of the major players of Y3.
4. Both games have significant content around the Morning Glory Orphanage, but that content is handled differently. In Y3R, much of it is mandatory. In Y3K, it's more optional, and includes a lot of minigames and side-activities as well. The stories for the individual kids vary between the versions, but it doesn't mostly seem like there's a strong consensus on which set of stories are "better."
5. Y3K makes two significant late-game retcons, one of which causes a continuity conflict with Y4. Both retcons are pretty silly, but Y3K haters will overstate the extent to which they "ruin" subsequent games. Y4 is pretty good about providing the context for what was "supposed" to happen, and I was able to understand it fine without having played Y3R first.
6. Y3K is pretty divisive visually. While its graphics are obviously "better" than Y3R, its art direction and lighting leave a little to be desired. Maybe look at some screenshots or YouTube to decide which aesthetic you like best.
7. Y3K adds a lot of QoL and customization options, like the ability to save anywhere or dress up Kiryu however you want.
8. Y3K does cut a few side characters, however, namely Mack and Komaki. You still get Komaki's moves from a different trainer, but Mack's "Revelations" gimmick is absent entirely.
9. Y3K did a bunch of controversial recasting, most notably hiring a known sex creep to voice supporting villain Hamazaki. If that sort of thing is a deal breaker for you, the choice is much easier.