How do I differentiate between new and old Titanium Apple Cards?

Presumably, the new card replaced one that is no longer valid, and you no longer with to use it. Is that correct? There is no way to distinguish between the two visually, other than to mark them in a manner that makes sense to you.

How to request or replace a titanium Apple Card - Apple Support

If you no longer need the old one you can send it to Apple for recycling. Start here: Apple Trade In - Apple, select Recycling, Get Started, then select Apple Card.

The old card becomes deactivated, so no one can use it anyway.

You can’t. They’re physically identical. Once the new card is activated, the old card is deactivated and can’t be used to complete transactions.

You could always try purchasing something and one will work and one won’t.