A bridge loan lets you borrow against your current home's equity by creating a new lien, giving you the funds to buy your new house. The bridge loan gets paid back when you sell your current home.
Recasting allows you to borrow more on a new house now without tapping your current home's equity. Simply put less down on the new house; then apply the equity from your current home (after it sells) to the new mortgage, lowering your loan balance and monthly payments.