There are several difference between the two. First, though, one similarity is that in drama (in theater), "comedy of manners" and "sentimental comedy" both refer to stage productions intended to acted, not to novels (as novels such as Jane Austen's are called comedy of manners as well). The differences between these types of comedy are several yet both are derived from Aristotle's definition in Poetics of comedy. In brief the definition stipulates that comedy is "a picture of the frailties of the lower part of mankind, ... since low life and middle life are entirely [comedy's] object" (Oliver Goldsmith, "A Comparison between Laughing and Sentimental Comedy" 1773).
Comedy of manners is meant to elicit laughter or at the least elicit amusement, often using satire as a means to its ends, as in the Restoration era works of Oscar Wilde and George Bernard Shaw. Sentimental comedy is meant to elicit tears and is called "weeping" comedy by Oliver Goldsmith in the 18th century: "The weeping sentimental comedy, so much in fashion at present" (Goldsmith).
Comedy of manners tells of the failings and follies of upper class persons who encounter moral dilemmas of various magnitudes and must or do and do not work their way out of them successfully. Sentimental comedy tells of the moral struggles of middle class people who are inherently good but led astray by bad example.
Comedy of manners questions and exposes contemporary society and social values by focusing on immorality particularly in romantic liaisons. Sentimental comedy presents moral values and asserts the power of noble qualities in overcoming trials and recovering virtue. The first concerns itself with society and social values, like in Wilde's The Importance of Being Ernest, while the second concerns itself with moral choices, nobility of inner qualities, virtue, and reformation of misguided behavior.