Verifying a payment method is telling everyone that you have a payment method such as PayPal, Skrill, and credit card, whereas authenticating your credit card is proving that the credit card you used to verify is yours.
Payment Method Verification
Verifying your payment method is simply adding a payment source to your Freelancer.com account. You can verify your PayPal or Skrill account, or credit card. This can be done through the Payments & Financials section of your Settings page.
You can also hover over your balance from your Dashboard and select Verify Payment Method.
To learn more about how to add a payment source to your account, you can read this article: How to Verify Your Payment Method.
Credit Card Authentication
Credit card authentication, on the other hand, is the process of validating the ownership of your credit card account/s on your Freelancer.com account. This is done through the Trust & Verification section of your Settings page to confirm that you own the card/s that you verify, thus, ensuring the safety of your account.
To learn more about authenticating your credit card and the detailed steps on how you can do this on your account, please read How to Authenticate Your Credit Card.