Update:
Sony offered up some more info on Yakuza Kiwami over on the PS Blog, the highlights of which you can find below.

All the in-game graphics were redone for PS4. And I?m not talking about doing up-rezzes of existing art, oh no. Graphics were completely redone for the 1080p/60 frames per second PS4 environment, and boy does it show.
The audio lines were re-recorded by the original cast (in Japanese). Astute fans may recall the US version of Yakuza being the only Yakuza game with a well-intentioned English dub, but given the complexity and uniqueness of the yakuza dialect, Kiwami exclusively has the original Japanese voices, re-recorded with the experience of the many Yakuza games since 2005!
You like combat" Boom! The combat in Yakuza Kiwami has been upgraded to the fast and (For legal purposes I can?t include the word I want to here, but it starts with an f and rhymes with ?curious?) combat style of Yakuza 0, where Kiryu can change stances on the fly to chain together combos and pull off the outrageously satisfying Heat Actions the game is known for! Substories! Finding substories (side missions) has become commonplace in the generations since Yakuza?s release, but were a bit obscure or difficult to find back in 2005. That?s been fully revamped now, with a clear delineation between the critical path missions and some of the quirky off-shoots of the main story.
Majima Everywhere. This is literally the best thing ever. Majima, the one-eyed, mad-as-a-hatter psychop...