Even though it is highly improbable I think there are actually two questions to this:

a) what happens if the same address (public key) gets created
b) what happens if the same private key gets created

a)
As far as I know a certain address usually holds only parts of your wallets Bitcoins (worst case scenario all),
so if the same address gets created but the private keys are different only the amount assigned to that address can be spent by both people (because their different private keys are locked with the same address).

b)
If the private keys are identical all the funds of each wallet could be spent by each user.

(What happens if both have different amounts of BTC in their wallets - what would show on the client)

Or did I get something wrong here?