Both discriminative and generative models have the ability to learn model parameters in a classification setting, but they are based on entirely different mechanisms and serve different purposes.
Pros: Well-designed for multiclass problems
Cons: Assumes multivariate normality among features, though it is relatively robust to this assumption
Both QDA and GMM are based on the premise that the underlying data is generated from two or more Gaussian distributions, but QDA is usually used in the supervised learning context, while GMM is used in the unsupervised context.
LDA assumes heterogeneity among class variances, while QDA allows for each class to have its own variance.
Discriminant analysis is a dimension reduction approach similar to principal components analysis but applied in a classification context.