There are two approaches to take in defining "wit." One approach is to define it as it used today. The typical current definition of "wit" is:
wit: intellectually amusing language that surprises and delights (Stanford University Literary Terms Glosary)
The second approach is to examine the use of "wit" throughout the literary ages because the meaning of wit has undergone several changes over time; Kip Wheeler, Ph.D., Carson-Newman College, gives a neat short summary of the changes in Literary Terms and Definitions.
Root: it comes from the Anglo-Saxon word "to know."
Renaissance writers Chaucer, Sidney, Spenser, Shakespeare: wit is the ability to write poetry with great intellectual skill and language mastery; thus Shakespeare was a wit.
Jacobean writers Jonson, Donne, Hobbes: wit is more specifically defined as "intellectual originality, ingenuity, and mental acuity" (Wheeler); thus Donne, with his original metaphysical conceits, was a great wit.
Enlightenment era figures Dryden, Pope: wit was putting old, established ideas into new and surprising words that created new and surprising images, including paradox: wit becomes more of a craftsmanship skill than an innate quality of mind.
Romantic poets Wordsworth and Coleridge: defining "wit" turned into a great debate in which "imagination" overshadowed "wit" as being the superior expression of mental powers; "imagination" was seen as superior because it was seen as the source of inspiration and understanding while wit was merely verbal gymnastics. Nonetheless for writers like Austen and Scott, "wit" in social settings remained the intellectual ability to amuse through the intelligent combination of ideas and paradoxes.
Victorian Matthew Arnold: wit became a low comedic devise that had no comparison to "imagination," thus Arnold condemned Chaucer and other great historic wits, like Jonson and Pope.
Modernist T. S. Elliot re-examined wit and gave it restored value in the hands of authors who combined intellectual humor with seriousness of poetic skill and profound themes.
Thus, "wit" is hard to define for a reason as it has undergone many subtle changes, thus knowing how to correctly define and understand "wit" requires specifying which period of literature you are considering.
"Sensibility" is defined by Stanford as the reliance "on feelings for truth." In other words, characters like Marianne in Sense and Sensibility found truth in relationships and truth in correct action through their feelings. As a result, Marianne wept uncontrollably for a protracted time for her father's death and determined that Willoughby would forever be her loyal and devoted true love based solely upon her feelings. Of course, Willoughby proved her philosophy to be wrong. In short, "sensibility" "values emotionalism over rationalism" (Wheeler).
Wit is mental sharpness and inventiveness, keen intelligence, or a natural aptitude for using words and ideas in a quick and inventive way to create humor. Sensibility is the ability to appreciate and respond to complex emotional or aesthetic influences or a person's delicate sensitivity that makes them readily offended or shocked.