Yes, the PUA or UC preceding the 1099-G form name is used to report different types of unemployment benefits received but do not change the way it is reported on your tax return. It is for reporting unemployment compensation received.
When you file your federal income taxes this amount must be reported on your Tax Form 1040 or 1040A.
If you received UC-1099G and PUA-1099G forms the amounts will have to added together before reporting on your tax forms.
If the box numbers don't match then go by the description for each box when inputting in TurboTax.
Here is a link with additional information:
Different types of Form 1099-G for Unemployment Benefits
