quantum mechanics - How are LS and JJ coupling different if both involve just vector addition?

In LS coupling the orbital angular momenta of particles $L_i$ couple together to form L. Similarly the spin angular momenta $S_i$ separately couple together to form S. Then S and L are coupled to get J.

In JJ coupling the $L_i$ and $S_i$ of each particle is coupled first and the resultant $J_i$s then combine to form J.

From the above, LS and JJ differ in the order of combining those vectors. If coupling is vector addition of the momenta (an associative operation) then how can it depend on the order of addition?