According to Jake Fuentes at Medium, the protests allow the Trump administration to purposely test the limits of governmental checks and balances for the sole self-serving purpose of consolidating power. From Trump's perspective, he gets to test the system by doing what he wants, thus getting exactly what he wants. Fuentes suggests that the administration never planned to actually use the green card element of the ban.
Protests do make a difference, as Slate's Jammel Bouie shows by looking at the recent Women's March. Its nationwide and global unity against the Trump administration flustered the new president, causing him to rush into making wrong decisions. Bouie further makes the point that now that Trump is president, it's a whole new ballgame because the backlash against him can hurt the entire GOP as mid-term elections come into focus.