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
X3/Y = Fathers
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!
There was actually a 1960s TV sitcom based on the notion of identical cousins. It was called the Patty Duke show. In reality the only way to have cousins that look alike would be through a quaternary marriage. That is when twins marry other twins. The children from the two marriages would technically be 1st cousins yet genetically they would be siblings as well. In other words, it would be impossible to figure out who the biological parents are based on the DNA alone since the parents (being identical twins) would share the exact same genetic material. The truth is the Patty Duke show scenario is highly unlikely. Patty and Cathy could not be identical cousins because the only way we get identical offspring is when a fertilised egg splits into two cells. Those identical cells grow into identical twins. So there would be a strong resemblance between Patty and Cathy but the odds of them looking like identical cousins is unlikely. More likely they would look like fraternal twins. Patty would indeed look like Cathy but you could tell the difference between them just as you could between any pair of sisters. The cousins wouldn’t look identical but they wouldn’t look like cousins. They would look like sisters.