I own Katamari Damacy and We Love Katamari. When Beautiful Katamari Xbox 360 version was released, I choose to rent the game first. This was a good decision.
Beautiful Katamari has some similar traits to the previous versions and the gameplay is basically the same. The game will store your information about cousins, presents, and items that you have collected. Similarly to Katamari Damacy the menu screen is featured in outer space and you can fly from planet to planet. This area is called "Princedom".
The graphics are just about the same quality to the PS2 games. Beautiful Katamari does have a higher resolution and more visible items. However, some items and background scenes have not very clear and appear blurry. Gameplay does stagger and have slowdowns. Controls and moves are the same, there are no new controls to learn.
There are a few new features added like achievement points (which have no real game value). A co-op mode and versus battle mode are available, but with only three levels. I would not suggest buying this game for these features, they are a total let down and will have you say "WHAT?"
Since Beautiful Katamari is for the Xbox 360, you can play using the Xbox Live versus battle mode. You will have the same concept of planets, which would be like Princedom. There are additional options like, choosing your opponents, area, battle, and signal strength.
An interesting twist to the Xbox Live feature is the leaderboards. For each battle, players are awarded cookies for their collecting skills and rolling. Just like achievement points, the cookies have no real value, but you can compare your stats with other players.
I would suggest renting Beautiful Katamari before purchasing the game. If you prefer a single player game action, you may just complete the game in a day or so. However, after renting the game and you find the Xbox Live feature is worth the price - then go ahead and buy it.