"She had something to suffer perhaps when they came into contact again, in seeing Anne restored to the rights of seniority, and the mistress of a very pretty landaulette"

Landaulette carriages were a kind of Regency sports coupe. Compact and stylish, they held half the capacity of a barouche landau (seen here in the photo). 

Creative Commons Attribution Share Alikelandau - Credit: Henry Salome

Anne Wentworth would have been able to travel in style (much to the envy of her younger sister Mary).  Her rights as older sister, and owner of a fashionable vehicle, would have, in a less steady young woman, been heady indeed, but Anne is made of more intelligent stuff.