In multivariate regression there are more than one dependent variable with different variances (or distributions). The predictor variables may be more than one or multiple. So it is may be a multiple regression with a matrix of dependent variables, i. e. multiple variances. 
But when we say multiple regression, we mean only one dependent variable with a single distribution or variance. The predictor variables are more than one. 
To summarise multiple refers to more than one predictor variables but multivariate refers to more than one dependent variables.