Deterministic wallet means that all of the keys/addresses that are used by the wallet are generated from the wallet's initial seed. (As opposed to the older classic style where 100 keys are pre-generated randomly, and more keys are randomly generated as needed.) The benefit of a deterministic wallet is that a single backup is enough to cover all of the funds that will ever be kept in the wallet. (In the classic wallet style, if you don't make a new backup regularly every 100 addresses, then funds in later addresses won't be accessible through the backup.)

Also - it's a perfect non-electronic way to have your funds.

Seed could be a sequence of words (this thing needs to be backed up and stored safe!)

Huge advantage is that you can save the seed in safety deposit box in a bank, written on paper - totally non electronic way.

Then you can recreate wallet from seed at any time and manage your funds.

Sorry, but how is that any more non-electronic than storing your ATM PIN in a safe? Given that the weakest security link is invariably human.

The ATM situation has a ton of links including your bank and the possible records of everywhere you've used your debit card.

If your bitcoin private key only exists on a piece of paper and not on any networked computers, then there's no way that malware or a company being hacked or an exchange going down is going to affect its balance.