Elinor and Marianne represent, respectively, the sense and sensibility of the title. Their biggest contrast lies in their respective temperaments, which really couldn't be any more different. Elinor is a cool, rational, sensible young woman, someone with a strength of understanding that allows her to offer wise counsel to her mother, even at so young an age. She is considerably more mature than her mother and prevents her from acting imprudently. Though still an affectionate young woman, Elinor is able to keep her emotions firmly in check.
This is more than can be said for Marianne. As the epitome of "sensibility" (a word that in Austen's time connoted emotional intensity), she is driven by her emotions to an alarming extent. Entirely lacking in self-restraint or the ability to control her feelings, Marianne is especially vulnerable to emotional hurt. She insists on following her heart, wherever it might lead, even when a moment's thought would've told her that it will inevitably all end in tears. That's precisely what happens when Marianne falls for the dashing but dastardly Willoughby, who cynically preys upon Marianne's emotional immaturity before casually and heartlessly discarding her.