Difference between spin-orbit coupling and LS coupling (Russell-Saunders)

Basically, in atomic physics, you would have two electrons, each with an angular momentum $l_1$ and $l_2$ and spin $s_1$ and $s_2$, and you want to couple all those to get the best approximation for the resulting spectrum.

So you have two options:

1- You couple $l_1$ and $l_2$ to $L$, and $s_1$ and $s_2$ to $S$, and then you couple $L$ with $S$ to get $LS$.

2- You couple $l_1$ and $s_1$ to $j_1$, and $l_2$ and $s_2$ to $j_2$, and then couple $j_1$ and $j_2$ to to get $JJ$.

So you have two ways to couple those, and the choice depends on how far the electrons are from each other where the specific angular momentum coupling is more pronounced. So if the electrons are close to each other, then you use LS coupling. While if you have them far apart, you use JJ coupling.

I hope this helps.