When identical twins marry identical twins, the children of the couples will be full genetic siblings but why are they not genetically identical?
They are not identical for the same reason full siblings are not identical.
A simplistic Mendelian setup would be the following:
X1/X2 = Mothers
That means the children could be:
So children born to two married couples composed of identical twins only have a 50/50 shot at being the same gender, much less have the exact same set of genes and express them in the same way.
Now, the simplistic view above only concerns 4 alleles - but the human genome is composed of thousands. By the end, the chances of genetically identical offspring from a set of identical twins is still very, very low.
And you haven't even began to include epigenetic variations, which often arise as a response to the environment! One cousin might be shorter due to their diet as an infant, one might have allergies the other does not depending on if they played outside as a child, and for non-Mendelian traits (incomplete dominance, co-dominance, phenotypes that rely on more than one allele [especially hair, eye, and skin color]) there's even more opportunity to introduce differences!