Marvel Studios marketed the title of the movie as Thunderbolts* but are referred to in the movie as The New Avengers. They even updated posters and other materials to resemble the new name. Although it was used to generate buzz and engagement with the movie, I do not think it was an accurate description of who the Thunderbolts are.
The Thunderbolts movie takes place after the events of Captain America: Brave New World and is set in the Year 2027. The movie is meant to pave the way for Avengers: Doomsday and Secret Wars but still be a standalone movie.
The Thunderbolts are a group of people assembled by Valentina Allegra de Fontaine (Julia Louis Dreyfus). Originally, she tricked them all into going to a secret O.X.E. bunker and attempted to incinerate them. Despite this they worked together and managed to escape. They are not superheroes in the traditional sense; they are people with complex backstories and motivations.
Calling them “The New Avengers “makes them look less than the original Avengers. If they are really meant to be the people in charge of protecting the world after the fall of the others, they need to have their own identity. Being constantly compared will not give them the confidence in themselves nor will it give others confidence in them. This would make their jobs even more difficult.
Whereas giving them a separate name like “Thunderbolts” depicts change; a new era with new superheroes who still want to protect innocent people.
Changing the name of Thunderbolts helps Marvel Studios but will not help the actual Thunderbolts or their reputation.