After enabling the State & Country as Picklist, when you query one of the new picklist fields, such as Account BillingCountryCode, which is the newly created Billing Country picklist on the Account, it returns the ISO Code value to you. So if the field shows "United States" through the UI, you get "US" back in the query of that field.
When trying to update that field (BillingCountryCode), or any of the standardized picklist Code fields, you must use the 2-letter ISO Code values to make the update.
It would be helpful if we could also use 3 digit ISO code values (e.g. "USA" for United States), as well as accept a full text value that matches the picklist value (e.g. be able to accept "United States", since that is the exact picklist value).