Most people won't even need to know the public/private key exists, but techies can leverage it to automate their lives some day.
Let's say I want to open a bank account with bank X. So I go to bank X's website and click open an account. Select my country and it redirects me to an oauth login page for my government. I log in and authorize the bank to view my basic identity info. Now the bank knows who I am and I can finish opening the account.
For in-person interactions, I could have my phone scan some person/company's public key in the form of a QR code and generate a qr code of my own that will let that company instantly authenticate me (airline, cruise line, police officer, etc.) with the government, but that same qr code will not do anything with any other person who sees my code because it is encrypted with the company's public key.
My government password can be reset if compromised and I never have to give out to anybody which puts it miles ahead of social security numbers from day 1. My phone is secure if lost, because it requires my government password to unlock the app as well as some biometric information.