Does Kiwami offer more than simply upgraded graphics? What about the combat? Or just the gameplay in general? Is it more Super Mario All-Stars or Resident Evil REmake?
More like REmake.
You get:
New story scenes and a few existing scenes are extended (like the first appearance of Majima in chapter 1).
Speaking of Majima, the Majima anywhere system means you'll meet him loads of times in Kiwami (in some rather strange encounters) to power up your Dragon of Dojima fighting style.
Combat is lifted from Yakuza 0. So the combat is way more fluid compared to the original release of Yakuza 1, you get more moves, more fighting styles (you get all 4 fighting styles from Yakuza 0).
Generally, the game controls better - especially for combat. Lock on works better in this game, and the camera is far more reliable.
Many more mini games to play - Yakuza 1 only had a couple of mini games, which this one includes far more to enjoy, including Pocket Fighter Racing tournaments, bowling, karaoke, Mah Jong & Shogi. Basically all the main stay mini games. Arcade kinda sucks though compared to how great it is in 0 (at least 6 fixed that).
More side missions (mainly tied to the new play spots added) plus a couple that tie into 0.
So basically they take 0 as a base, then chuck 1's story into the game (while fleshing out a few character arcs while they are at it).