While 90% of statistics reported in the literature fall under the guise of between-subjects and within-subjects analyses, they do not properly account for all of the variance and confounding effects that exist in reality. Multivariate statistics play an important role in empirical reasoning because they allow us to control for various demographic, confounding, clinical, or prognostic variables that mitigate, mediate, and affect the association between a predictor and outcome variable. They are also much more representative of reality and true effects that exist within human populations.
Very few if any relationships or treatment effects in physiology, psychology, education, or life in general are bivariate in nature. Relationships and treatment effects in reality ARE multivariate, diverse, and confounded by any number of characteristics. Therefore, it makes sense that researchers should be conducting multivariate statistics to truly understand human phenomena.
With this being said, it is important to use multivariate statistics ONLY when you are asking a multivariate research question. Throwing a bunch of variables into a model without some sort of theoretical or conceptual reason for including them can yield false treatment effects and increase Type I errors. Also, these spurious variables can create "statistical noise" which detracts from a model's capability for detecting significant associations.
Choosing the correct multivariate statistic to answer your question is simple. You choose the multivariate analysis based on the outcome.
1. Categorical outcomes - Logistic regression (dichotomous), multinomial logistic regression (polychotomous), Kaplan-Meier, Cochran-Mantel-Haenszel, Cox regression (dichotomous/survival/time-to-event)
2. Ordinal outcomes - Proportional odds regression
3. Continuous outcomes - Factorial ANOVA with fixed effects, factorial ANOVA with random effects, factorial ANOVA with mixed effects, ANCOVA, multiple regression, MANOVA, MANCOVA
4. Count outcomes - Negative binomial regression (variance larger than mean) and Poisson regression (mean larger than variance)