In general I think that opening sequences are important because they can display a lot about the movie without the audience knowing that they are learning about the movie. For instance; in all the James Bond movies like Skyfall, what happens during the movie is displayed in the title sequence whist the credits are being shown, which makes the audience think it is just something to entertain them whilst the credits are on, which it is but it is mainly to show the storyline of the movie.
Also the title sequence will set the mood and tone of the audience for the rest of the movie. This is mainly seen in horror movies because the sound used causes the audience to become tense and then they will still be tense from the title sequence by the time the movie starts so they are in the right frame of mind; otherwise if the tension wasn't built up in the title sequence then when the film started they wouldn't be as scared as they are with the help of title sequence.
The title sequence can display the main location of the movie like in 'Forrest Gump' because it tracks a feather but it discreetly shows the setting of the film and this leads to the fact that title sequences display who the main actor of the film is by showing their face before you see anybody else's.
The main importance of title sequences is to display to the audience who the main actors are in the movie and who the director is and other people involved in the production of the film.