Tywin loved Joanna, true. I also think he cared for Kevan.
I think he saw his children as his legacy and tried tirelessly to mold them into what he saw as best for the Lannister family. They were there to carry on what he had started, and he was most dismayed when they acted on their own.
I think he loved Jaime but was flabbergasted by what he saw as his son throwing away all that he had worked to give him.
Cersei was a girl and therefore did not really make a blip on her father's radar other than as a potential pawn.
He obviously loathed Tyrion for his form and "killing" Joanna.
I think the show did a good job of showing who Tywin is in those scenes with Arya. (I waffle on whether or not Arya as a character suffers as a result. She loses some of the "kill" scenes, and I am inclined to think that a good thing. Martin made her almost too bloody.)
Cersei for all her flaws did love her children, but her love blinded her and made her a terrible mother.
She never allowed herself to see what Joffrey was and then cannonized him on his death. She then used Joffrey's death to berate poor Tommen for not being like his brother. She thought of Joffrey was strong when he was really snivelling and deranged.
Kevan and even Jaime eventually had the same thought. Tommen would be better off removed from his mother's influence. Tommen is sensible and not unkind. He could be molded into a good leader.
Cersei learned few things from Tywin, but she learned only his hardest lessons. She saw how his cruelty won him certain things but never learned that too much cruelty and sheer spite can be as damaging as being too soft.
Even without the ear (Thank you Dark Star), Myrcella is far better off in Dorne.
In the show they have taken one of her earliest and most heinous offenses (Robert's illegimate childrens' deaths) and tacked it on to Joffrey. I don't blame them much for that, but Cersei's descent into madness is compelling in the books.
At least I thought she was going mad. I gave her much more credit for cunning and game playing before Martin let us in her head. Then she became this paranoid small creature. I am still not sure how I feel about that.