Editions: Daedalus Rewards
Daedalus wallet refers to the application that you may have downloaded previously. It works on Mainnet and has real ada.
Daedalus Rewards refers to the application you need to download and use to do the balance check - to compare funds between Mainnet (Daedalus wallet) to Incentivized Testnet (Daedalus Rewards).
Wallets within the application
Balance wallet refers to a wallet that you created previously on mainnet and that was captured in the snapshot. The words Balance wallet are only visible in Daedalus Rewards not in Daedalus wallet on mainnet. You restore into Daedalus Rewards using your 12-word recovery phrase from Daedalus wallet created originally on mainnet, or using your 27-word recovery phrase from Daedalus Paper Wallet created originally on mainnet. You cannot do much with the Balance wallet except transfer your funds to a rewards wallet using the balance transfer wizard that will pop up after you have restored. You cannot stake or earn rewards from this type of wallet.
Rewards wallet refers to the wallet that you can create in the Daedalus Rewards wallet to use on Shelley Incentivized Testnet, it has a 15-word recovery passphrase. You can use a Rewards wallet for staking on the Shelley Incentivised Testnet once staking is available. You can also use it to send and receive funds on the Shelley Incentivised Testnet, including receiving funds from a Balance wallet.