The problem was, that transactions can have "change". So by sending 1 BTC, the client did in fact send 1 BTC to one address and the remaining 8999 BTC to a a newly generated address which is automatically added to the wallet. Since the backup was before this transaction, the backup didn't include the private key to the change address and 8999 BTC were lost.
A really unfortunate incident, as the person simply didn't know this technical detail. It prompted some upgrades to the client, which now generates addresses in advance and stores them in the wallet. When new addresses are needed, the ones from this pool are used first. With a recent client this accident would not have happened. It's still important to know though, that after _sending_ Bitcoins, a wallet might potentially include new private keys that are not in an old backup (if the pool has been completely used up since the last backup).
By the way: All this happened in August 2010, so this wasn't $81,000 at the time.