Rome is a wonderful city with so much to see. It is also quite tiring - especially in hot weather. It has been my experience that it is worth paying for a really central hotel - ie near to Piazza Navona, Pantheon, Trevi Fountain. From this area you can walk to anywhere and you can also pop back to the hotel for a siesta, change shoes etc during the day & again before dinner.
If you are away from the centre, the hotel may be cheaper but you will spend more on transport & will probably have to stay out all day because of the cost & time taken getting back to the hotel.
The Vatican is not central and if you stay there you are quite a distance from anything else in Rome. Better to be in the middle of everything.