Compare and contrast the characters of Sophia and Olivia.

To a large extent, both Olivia and Sophia reflect the characteristics of their father. They are fundamentally kind, decent, and respectable people, but they have their faults and are prone to making foolish decisions. Olivia is an extraordinarily vain social butterfly, obsessed with her looks and appearance. She also has an unhealthy fixation with social status. As far as she is concerned, she is so much grander than her family's circumstances would suggest. Sophia, though also concerned with her place in society, is a much more gentle soul.
Olivia is very much a creature of both her time and her family. Despite coming down in the world, the members of the Primrose family take great pains to portray themselves as thoroughly genteel. Eighteenth-century England had a much more hierarchical society than we do today, and social status was all-important. It is not altogether surprising that Dr. Primrose's stubborn refusal to accept his lowly position in life leads to his enthusiastic endorsement of Squire Thornhill's courtship of his daughter—despite Thornhill being an improvident rake and libertine.
Sophia is much less restricted by social conventions than her sister. She is possessed by a romantic streak and does not see marriage in the terms of a glorified business transaction as her family tends to do. As a result, we are not terribly surprised when she falls for the dashing Mr. Burchell, even though he is apparently poorer than the Primroses themselves. That being said, she lacks the impetuosity of Olivia. We certainly cannot imagine her eloping with anyone as Olivia does with Thornhill.
Olivia and Sophia are very interesting characters because they each represent a particular facet of their family's self-image. Olivia represents the status-conscious side of the Primroses, striving with all her might to become someone;  she wishes to become a refined, respectable lady at the very pinnacle of rural society. Sophia, on the other hand, represents another side altogether; she represents a virtuous side, more moral and more intelligent, with a genuine understanding for what is right and proper.