As the title suggests, Austen's main point in Sense and Sensibility is to philosophize or moralize about two different perspectives or personality types. Hence, since Austen uses two sisters to make her point, the main conflict in the book is actually twofold. The first main conflict in the story is between the two sisters. Elinor rightly judges Marianne as being too irrationally emotional, or relying too much on her sensibilities, while Marianne incorrectly judges Elinor as having no heart. The second main conflict is that Marianne finds herself in conflict with herself because all of her romantic ideals lead her to not only fall in love with a man of ignoble character but also to destroy herself, nearly causing her death. Hence, the first main conflict is character vs. self.
When Marianne first meets Willoughby, both she and the reader believe that he is the ideal man. When Marianne slips and falls, twisting her ankle, while walking through the downs with Margaret, Willoughby appears out of nowhere in the rain and carries her home safely. He is handsome, charming, and gallant. As Marianne phrases it, "His person and air were equal to what her fancy had ever drawn for the hero of a favourite story" (Ch. 9). However, we soon learn that his character is actually far less than ideal. Not only is he extravagant, we later learn that he is guilty of seducing and impregnating Colonel Brandon's charge, Eliza. Marianne's heartbreak over Willoughby lead to a raging fever that nearly takes her life, emphasizing the character vs. self conflict. However, eventually Marianne realizes that, like her sister, she could have born her heartache with more "fortitude"; she further realizes that her passionate, uncontrolled feelings led her to be selfish and hurtful towards others, especially Elinor. Marianne instead later resolves that "[her] feelings shall be governed and [her] temper improved," leading to the resolution of the conflict (Ch. 46).
Similarly to Marianne, Elinor also suffers a broken heart when she learns that the man she has fallen in love with is secretly engaged to Lucy Steele. However, unlike Marianne, she does not let others see the nature of her sorrow, one reason being that she swore to Lucy that she would keep their engagement a secret. Nonetheless, Elinor must remain the strong one. When Marianne is crushed by Willoughby, Elinor must comfort her; when Marianne learns of Edwards engagement, again, Elinor must comfort her. When Marianne asks Elinor how she bore it all, Elinor explains that she would rather selflessly spare anyone from knowing how she feels and making them feel equal pain. Marianne believes that Elinor really didn't feel much for Edward. But as Elinor argues:
Then, if I had not been bound to silence, perhaps nothing could have kept me entirely--not even what I owed to my dearest friends--from openly shewing that I was very unhappy. (Ch. 37)
Hence, we see that the nature of the character vs. character conflict between sisters is that Marianne believes she should be self-serving towards her emotions while Elinor believes that emotions should be controlled and that one must be selfless and pay attention to duty at all times. This conflict is resolved when Marianne sees the error of her ways.