Both Madame Aubain and Felicite are opposites for a variety of reasons.  The most evident reason is because of class.  Madame Aubain is a wealthy woman who owns property and has money at her disposal while Felicite is a servant, her servant.  Madame Aubain is educated and represents a construct of power, while Felicite lacks a certain amount of power to be able to determine her own condition and is not a proficient reader.  Madame Aubain possesses this ability to freely define her own state of being, while Felicite does not readily embody this condition.  I would say that another difference is that Felicite has a much more constructive view of life than Madame Aubain.  Cloistered in her world of wealth and privilege, Madame Aubain is one who is able to feel only through these visages.  She does not display herself as one who is able to live life and understand from it in terms of being able to experience all that life has to offer. Felicite, on the other hand, is much more adept to being able to learning from live and understanding from it. For this reason, Felicite's heart is so much more geared towards the demonstration and experience of love than Madame Aubain is.  Perhaps for this reason, she lives longer than Madame Aubain and lives more of a redemptive life than she does.

However, Flaubert ends up constructing both of them as more similar than different.  With the death of Virginie and the growing embrace of Felicite's Catholicism, Madame Aubain has a difficult time understanding the nature of death and life after it.  Felicite's religious fervor is what not only helps her through such a painful moment, but also helps Madame Aubain through it.  Both of them find one another in the search for meaning and understanding the reasoning in the death of a child.  It is here where their similiarities overwhelm their differences:

Their eyes met fixedly and filled with tears; at last the mistress opened her arms, the servant threw herself into them, and they embraced each other, satisfying their grief in a kiss that made them equal.

It was the first time in their lives, Mme. Aubain's nature not being expansive. Félicité was as grateful as though she had received a favor, and cherished her mistress from that moment with the devotion of an animal and a religious worship.

It is through love and the transcendental nature of religious worship where their differences are fused away.  In its place is a complementary relationship with Felicite possessing "the devotion of an animal" and Madame Aubain understanding the true nature of this woman who had been in her service for quite some time without even being realized how much a force of construction in a world of destruction that she truly is.