The difference in public and private speech covers political intent. Preparing to celebrate Caesar, Marullus decries the plebeians as hypocrites when he reminds them of how vigorously they celebrated Pompey when he was in power before Caesar ("Have you not made an universal shout/That Tiber trembled underneath her banks/ To hear the replication of your sounds/Made in her concave shores?"--You praised Pompey so loudly it echoed across the river! So why celebrate Caesar now?) Flavius follows up by "guilting" the crowd into not celebrating and dispersing. This is the first example of authorities using speech to manipulate the populace. However, the tribunes themselves are hypocrites--they are not so much pro-Pompey as their words suggest, but rather are anti-Caesar, and wish to "drive away the vulgar from the streets" to lessen the public show of support for Caesar. In their private speech immediately following the disbursement of the plebeians, they confirm this by planning to "disrobe the images" of Caesar as a means of protest, because they cannot publicly speak out against him; but for even this minor protest, they are killed ("put to silence") in I.ii, which not only confirms the conspirator's suspicions of Caesar's tyranny, but foreshadows the widespread bloodshed to occur when opposing and shifting political alliances have no means of resolving their differences, except by killing the opposition.
Yes, Act 1 Scene 1 is very important. Another issue concerning the tribunes opens here: How differently do the leaders of Rome behave in public compared to when they are alone together? The difference between what might be called "public and private appearance" is evident in the scene and will recur through the play. We see this when Murellus criticizes the crowd, reminding them of the greatness of Rome and what it means to be a Roman (lines 44-46), yet while doing so implies that the plebeians are citizens of Rome, with all the richness of heritage that carries. But when the crowd "vanish tongue-tied in their guiltiness" Flavius decides that they should set about removing the decorations from Caesar's statues, ending with a comment about Caesar's rise to power (71-74).