This depends on the wallet. They have to be able to "watch" addresses. Afaik bitcoin core does not support this yet. I know armory and blockchain do.
Electrum offers you to do the same - you need to copy over the Master Public Key from your offline client to your online one.
@OP: There is no difference - the website is updated with the newest version (although it would be better to download it from github so you can verify that it hasn't been altered).