Apple Pay versus Apple Wallet—what's the difference and what are common issues?

To me, the two seem to be synonymous. You can't add a card to your Apple Wallet unless it's Apple Pay compatible. However, I ran into an issue (an Apple senior advisor is looking into this) that things are not working as it should. For instance, recent transactions are no longer appearing in Apple Pay.

The last transactions registered were all in January, but I've made many Apple Wallet transactions to make NFC payments since then. I have about 6 different credit cards and debit cards stored in my Apple Wallet.

I also have issues in purchasing an Apple (monthly) subscription (Apple Care+) where my Apple account is not pulling any cards from my Wallet. I have to manually enter the card details (Add New Payment Method) despite it being active in my Apple Wallet. I also can't use my Apple Balance which was added via iTunes Cards as my default payment method despite being told I should be able to choose.

I can see no method of resetting my Apple Wallet short of wiping (erasing) my iPhone and setting it up as a new device.

Has anyone ran into my issues before?