Winter tyres and snow chains serve two completely distinct (albeit related) purposes.
Winter tyres give you more grip on normal wintery roads, may help if you come across slippery wet conditions or black ice, and in general make winter driving safer. They do not mean you can drive with less care than usual in difficult conditions, but they do mean it's safer to drive.
Snow chains, on the other hand, are for actual snowy and icy roads only. They need to be taken off and put on, and are a right royal pain (as some folk have mentioned).
You don't need one or the other, you need both; the tyres for normal wintery driving, the chains for when conditions become really difficult.
-simon