I never played the PS3/X360 version of DOA5LR; I just went straight through the next gen version. I never played Core Fighters either, and I don't think the X360 version had Core Fighters.
I am not sure. It's missing all the new stages such as Danger Zone and the last DLC it got was some Halloween ones, therefore I can guess it also doesn't have additional character changes.
Besides content, virtually nothing besides a new hair and misc. UI for the character select options and a skin shader slapped on and a few different particle effects. It looks better but nothing to write home about current gen wise
On PS3/360 version: it doesn't have The Crimson and Danger Zone stages, the customizer for hair options, glasses, etc, the skin shader and certain particle effects. Also due to limitations of DLC, it stopped right after the Halloween 2015 costumes.
Oh and it doesnt have the LR breast jiggle either. And if I recall correctly DOA5LR on last gen consoles hasn't received a balance patch since the Halloween costumes
Crimson and Danger Zone aren't the only missing stages. It's also missing Fireworks, AoT and Azuchi. The Mai and Naotora characters are also missing, as are several costume packs.
Additionally, the last gen version is not even a viable version of the game anymore, because they cut support part way through, so it isn't even running the same meta, due to being quite a few patches behind the current gen version.