I think Tywin's primary motivation is what people think of him and more importantly the legacy of House Lannister. Specifically, according to Kevan he mistrusts laughter and seldom smiled at all after the death of his father and wife Joanna. People mocking him or challenging his House's legitimacy drives him absolutely nuts.
Tywin also needs some bare level of legitimacy in the minds of all of the realm the great houses and even the commoners. Tywin does not care if they know he's ruthless as long as the end result is that all of Westeros knows of Lannister superiority. For example when Tywin ordered the barracks guards to gang rape Tysha and give her a silver stag each but Tyrion had to give her a gold dragon, "b/c a Lannister was worth more." Why does he care that one common girl like Tysha? I never viewed Tywin as unnecessarily cruel like Ramsay but this action seemed to waste his time, his guards, and his money for no reason than other to humiliate Tyrion and her. I think this goes back to his feelings about Tytos and his mistress who he made walk thru Lannisport naked.
However, Tywin is smart enough to realize he can't just directly commit atrocities across the realm. He needs just enough pretext or guise to be able to justify his actions or blame others. The Reyne and Tarbeck massacre while directly committed by Tywin, can be justified to some degree because of their rebellion. Although the entire house was put to the sword Tywin was still generally liked throughout the Westerlands. "The Rains of Castamere" is even a popular song so I think the whole realm somewhat accepts it.
He needs to have the nobility think there is at least a rational basis he didn't order the slaughter and he is on the winning side. While most of Westeros knows Clegane and Lorch scaled the Red Keep and killed Elia and the babes there is no actual direct evidence showing Tywin ordered it. .The fact of the quick Houses of Baratheon/Lannister uniting no one cared save Ned and Jon to condemn the actions. Aery's cruelty overshadowed any argument against Tywin and his lieges. Any accusations of Lannister involvement in the RW can be deflected to Freys too.
I think the scene before the Battle of the Green Fork Tywin sums it up nicely when he asks whether Jamie cares when people call him Kingslayer. Jamie says of course it bothers him, Tywin then says, "... a lion doesn't concern himself with the opinions of sheep." But maybe the lion should consider the opinions of the wolf and dragon regarding the sheep as well?