I would say that $\left\langle v_i \right\rangle\left\langle v_j \right\rangle$ is the product of averages, whereas $\left\langle v_i v_j \right\rangle$ is the average of a product. Hope you get the difference.
Take an example of $v_i = v_j = v$, then $\left\langle v_i \right\rangle\left\langle v_j \right\rangle = \left\langle v \right\rangle^2$, while $\left\langle v_i v_j \right\rangle = \left\langle v^2 \right\rangle$. It should be trivial now, that the quantities are not the same. lurscher's example also confirms this. You can find another example in statistics: consider a dataset of
v & 0 & 1 & 1.5 & 0.75 & 0.5
then $\left\langle v \right\rangle = (0+1+1.5+0.75+0.5)/5 = 0.95$, hence $\left\langle v \right\rangle^2 = 0.9025$, but $\left\langle v^2 \right\rangle = (0+1+2.25+0.5625+0.25)/5 = 0.8125$, which is by no mean equal to $0.9025$.
This was pure mathematical explanation, and there is no difference, which functions or statistical quantities you consider. Hope that now you will be able to explain yourself the physical difference of particular quantities you are working with.