When you become Wanted, you get a bounty placed on your head, and lose the ability to Give Up in a fight. The bounty is equal to 10,000 G times the week number. You also cannot land on any towns, castles, temples, shops or Dokapon Castle.
In the DX version, the Sheriff is notified, and will try to hunt down the Wanted player.
In the Kingdom version, the Accessory "Criminal Studs" will give you a temporary raise in stats, but only if you're Wanted, so if you successfully rob a shop, town or merchant, you will not get the raise in stats. The amount of days Wanted can also be reduced if you have the "Nice Guy Badge" Accessory. The Ninja end of week bonus involves Attacking/Robbing a Shop. Even if you fail to rob something, you will still get the bonus.