At the beginning of A New Hope, Leia is some kind of big-shot courier in the Rebellion. She's aboard the Tantive IV with the Death Star plans, evidently both as custodian of those plans and to provide the ship with consular (or higher) immunity. She seems to be the spokesperson for her father, a high muckity-muck in the Rebellion from the beginning; Alderaan probably supplies a lot of the funds and materiel for the Rebellion. She's also an Imperial Senator, giving her a certain amount of access on Coruscant. She's a little bit Benjamin Franklin, a little bit Benedict Arnold, and a little bit Jefferson Davis.
Fast-forward to the time of Empire Strikes Back. Alderaan has been destroyed, and the influence of its people in the Rebellion marginalized. There's an Alderaanian diaspora throughout the galaxy, but Leia isn't organizing that. (Note she retains her HRH style throughout the Trilogy; someone else in the Royal Family outranks her.) The Emperor has probably maneuvered one of his lackeys onto the Alderaanian throne. The Imperial Senate has been dissolved. Leia has been outed as a Rebel. She can no longer act as an open courier, no longer has access to most of Alderaan's power, no longer has access on Coruscant.
So why do they keep her around? She's got a pretty face, but she doesn't put out. She can shoot, but she doesn't go on missions. She confuses and angers the civilian contractors. My guess is, she's still valuable to the Alderaanian cells of the Alliance for propaganda reasons. That's why she's schlepped from Yavin IV to Echo Base. So she can make encouraging videotapes to be played on the Holonet equivalent of YouTube. Look at the warrior-princess soldiering on at the front lines. But she's not important to the Alliance as a whole; that's why she's not enjoying the good life with Mon Mothma on the Calamarian fleet.
Of course, things changed by Return of the Jedi. She was an undercover special forces operator. And Han and Lando were generals, and the Falcon was a starfighter, and so forth and so on. The less said about that, the better.