Such a great question! Clearly, Austen had a lot to say on these topics because she titled a book that way - but the novel "Emma" proves that she had so much to say it couldn't be just contained in one book!
In "Emma", Austen uses her characters to show these opposing traits, and ultimately makes a case (as she does in "Sense and Sensibility") that "sense" is best. Lets start with the trait of sensibility - that is, to behave and decision-make based on emotional concerns as opposed to intellectual reasoning.
Harriet is the most prone to sensibility. She is a sweet girl, but not very well-read or intellectual in her pursuits. She enjoys being with people and pursuing enjoyment in various forms - she is a pleaser, and wants to please others. The proof of her emotional tendencies lies in the ease with which her romantic attachments form and reform. She falls for Mr. Martin, but gives him up at the persuasion of her friend for Mr. Elton. She seeks to please Emma, and is ready to be swept up into love at the idea that Elton loves her. When that fails, she allows her gratitude towards Mr. Knightley to convince her that she is in love with him. She claims to have been swayed by his actions, but she is is painting the scene with her own bias and not considering all the implications.
Mr. Woodhouse is another great example of sensibility. Harriet has a tendency towards love - Woodhouse has a tendency towards fear. He fears the weather, change, and various other potential and made-up concerns. He "explains" his fears, but he will not be explained out of his fears. He must be tricked out of them, or he must be accomodated.
Frank Churchill is similar to Willoughby in the novel "Sense and Sensibility". He is concerned with enjoyment. He secretly engages because he does not want to endanger his own inheritance, and so puts enjoyment and comfort above honesty. In addition, he leads Emma on in a flirtation though he is not available. He even goes so far as to create a false scenario surrounding the piano - not because he needs to in order to divert suspicion, but because it is fun. He leads Emma astray slightly in this regard, which is important for her character development.
Knightley is the foil to these characters and the one who represents "sense" the most. He is always considering the pros and cons of a situation, always arguing for truth. He is the one to speak about the fitness of the match between Martin and Harriet, speaking specifically about finances. He is so reasonable and sensible that he must consult his brother before declaring intentions towards Emma.
Which brings us to the protagonist. Emma believes that she is full of sense, but she realizes in the climatic moment that she has been led by her sensibilities. This is when she insults Miss Bates, at the prodding of Frank Churchill. Knightley chides her for being disrespectful, and Emma must admit that he is right and that - as much as she "reasons" in her head about the world and people around her - she has been as emotional as her father in her ways of behaving.