Is there any point in the battle mansion you unlock after defeating the game? I've done about 8 battles, winning all of them with ease, and nothing seems to have come from it.
It's the most 'competitive' thing in the game other than battling online against other trainers. You need to do 20 straight wins on the regular version to unlock the 'Super' Mode for each kind of battle. All of the Pokémon in there will have perfect 'IVs' and the 'EVs' in the right stats to make them as strong as possible, the main aim is to defeat all the Leaders for each section, which in the Super Mode appear after 50 wins in a row. The Battle Points are good for getting competitive Items and certain useful TMs. You also get trophies that are displayed in the Maison's hall when you reach the 50 wins for each one.
Other than that, it's just about seeing how long you can keep your streak going. I know a Canadian player who's always looking to build teams that can go on massive streaks, his best in X & Y so far is over 1000! I haven't gotten around to much yet, I've just unlocked the Super Modes in Singles and Doubles, but the best I've managed is about 30~ on Doubles before I lost. I plan on building a team to do these soon.
You also get the 'achievements' for these. If you've used Game Sync to connect your game to your Trainer Club Account and PGL Account, you'll be able to see which achievements you've done in the game. This is another thing to aim for as there's at least 255 achievements to try and get.
Edit: Oh! There's also two kinds of Berries you get for getting to 100 and 200 wins respectively in any of the five Super formats. Only way to get these Berries so far as we can't send over Items from previous games.
Lastly, remember PokéBank is out on the 27th of this month on the Nintendo eShop. For anyone unaware, it allows you to store up to 3,000 Pokémon on a cloud service and costs £4.49 a year - youget a 30-day free trial though! So can easily abuse it without paying anything. Once you download the Bank, you get access to the Transporter App as well, which will let you send over anything you want from your Pokémon Black, White, Black 2, or White 2 versions to the cloud, and thus to X & Y as well (this is a one-way service, so you can't send them back to your DS versions once they're in the cloud or on X/Y). Even though the service only works with X, Y, Black, White, Black 2, and White 2, it still means you can send Pokémon over from as far back as Ruby and Sapphire (Emerald, FireRed, LeafGreen, Pokémon Colosseum, XD: Gale of Darkness) if you transfer them to your Diamond/Pearl/Platinum/HeartGold/SoulSilver and then to Black/White/Black2/White2. It's also anticipated that this service will work for all future Pokémon titles, so we'll always be able to cloud things from this Generation onwards.
If you don't plan on using PokéBank, at least download it along with the transporter to transfer all of your old Pokémon across and obtain Celebi. Then you're free to cut it and not use it any more after you've abused it :P