The same problem the Yellow Card [1] solves. Right now, I have a New York State electronic passport and a CDC paper proof of vaccination. The former works in New York City. The latter works across the U.S., at least now. (At least, it worked in San Francisco and Atlanta.) If I travel abroad, I may need to get the latter re-certified since it's trivial to forge and not independently verifiable. Having a single, authoritative document fixes all this and makes socializing and commerce more frictionless.
If we don't want to go into another lockdown when the Delta variant hits, we need to be able to isolate the vaccinated from the vulnerable. The former can continue mixing and mingling with reduced (though not zero) risk of causing a flare-up. The latter, I don't know, ideally they'd stay home but we know that won't happen.