The paper backup holds the key to generate all your private keys.
https://en.bitcoin.it/wiki/Deterministic_walletNo wallet holds bitcoins. The blockchain does this.
Normally clients get awfully confused if used with the same wallet on multiple computers.
Armory may behave a little better but consider your address book and transaction comments. These will not sync between your computers.
I recommend to use online Armory on one computer only.