Discriminant analysis's mechanics are different depending on the approach taken. There are two main types of discriminant analysis: Fisher's discriminant analysis and Mahalanobis's discriminant analysis. Because of the dissimilar processes involved in these two approaches to discriminant analysis, the resulting solutions are not alike. Add to this the addition of other, newer approaches to discriminant analysis such as Kernel discriminant analysis, and you will find that researchers using discriminant analysis yield inconsistent results on the same data sets. This is problematic when actually applying discriminant analysis's classification to real data points.
Due to the complexity of discriminant analysis's mechanics, it is an unwieldy tool for all those but the mathematically-savvy. Similar tools, such as multiple regression, are just as flexible as discriminant analysis without carrying along the intricacy and specificity associated with discriminant analysis. They thus are more popular with researchers. To truly understand the procedure of using discriminant analysis to produce a solution to a research problem, a researcher must have extensive experience with matrix algebra, matrix calculus and advanced statistics.
The prediction available through discriminant analysis is not true prediction. In statistics, prediction allows researchers to know certain properties of a data point to a very specific degree (such as within a confidence interval or margin of error). However, discriminant analysis does not have this characteristic; discriminant analysis instead classifies. In other words, discriminant analysis can only tell a researcher the likely grouping of a certain data point; it cannot tell the researcher other properties of the data point or how likely it is the data point is a member of the classified group.
While most statistical tools and models have easily applied goodness of fit tests, discriminant analysis remains a complicated beast. Instead of having easily computed statistics that can represent goodness of fit (e.g., R-squared for multiple regression), discriminant analysis usually uses "hit rates" to assess goodness of fit. Hit rate refers to how often discriminant analysis correctly classifies a data point. However, to even calculate a hit rate, you have to collect a new data set. Even upon doing so, the hit rate is problematic and inaccurate in the case where the group sizes are not equal -- which is almost always the case.