One of the advantages of visitting Santa Teresa is that you get too see the city center and Gloria from above. Also a visit to Tijuca national park has its advantages also due to the view.
The overall discussion, however, is whether it is worth it to visit a lot of places while in Rio or whether a visit to Rio de Janeiro should be a visit where you enjoy the beaches and to party.
In general, you can safely avoid visitting the city center. Rio de Janeiro has an interesting history, but not so interesting that you cannot have a 2 week vacation without a city tour. The buildings are not particularly beautiful and the few areas with some beauty you will see when visitting Santa Teresa.
Tijuca National Park gives a great view from the Chinese view, but beyond this there are not much wildlife in this area.
The surrounding cities are of some interest and can be worth a visit. However, if you travelled far to visit Rio de Janeiro and are only to stay for a little more than a week - it might not be the best use of your time.
In conclusion: After Sugarloaf and Christ the Redeemer statue, you can safely just try to relax and enjoy yourself without stressing about seeing the whole city.