The Delta Gold Amex only generates Delta miles. The Platinum, Gold Premier, and Gold Amex earn Membership Reward points that can be transferred to numerous frequent flyer and hotel programs (Delta and British Airways being the biggest partners since Continental left). Technically, the Green Amex earns MR points too, but they can't be transferred to airline miles unless you also have a Gold or Platinum account.

Since American Express dates all your accounts to the date you opened your oldest account, I would always keep one Amex open, or open a new Amex before you closed your only account. It will help your credit score.