As a Hilton Diamond member you get 20x from Hilton when you book direct regardless of your payment method on a paid stay (10x base plus 100% bonus from Diamond), and then if you pay with the Amex Aspire you get an additional 14x points from the card over and above what you get from Hilton.

If you paid for the Hilton stay on a card like the Bank of America Customized Cash Rewards w/ Platinum Honors status you could earn 5.25% cash back. Since Hilton regularly sells their points at 0.5c each that cash back could buy you 10.5 Hilton points. While this is not as good as the 14x from the Aspire card, it gives you the flexibility of cash to use that value somewhere else besides Hilton if you want.

If you're willing to lock yourself in to a program you can eek out a little more value.