Assume that $\vec X$ and $\vec Y$ are independent, nonnegative $d$-dimensional random vectors with distribution function (d.f.) $F(\vec x)$ and $G(\vec x)$, respectively. We are interested in estimates for the difference between the product and the convolution product of $F$ and $G$, i.e., \[ D(\vec x)=F(\vec x)G(\vec x)-F* G(\vec x). \] Related to $D(\vec x)$ is the difference $R(\vec x)$ between the tail of the convolution and the sum of the tails: \[ R(\vec x)=(1-F* G(\vec x))-(1-F(\vec x)+1-G(\vec x)). \] We obtain asymptotic inequalities and asymptotic equalities for $D(\vec x)$ and $R(\vec x)$. The results are multivariate analogues of univariate results obtained by several authors before.