The Derby is their entire season. Apart from obsessing about us, it's all they have. I agree with Neville that the Derby is over-hyped in this city, though. Mind you, that seems to be more from their side and the local media than from our side. Personally I couldn't give a crap about the Derby. It died years ago for me. The blue moronic element wrecked it in the 90s and it's slid further into the cesspit since.
I agree about the late 90's being the turning point for the Derby match and how it felt. To be fair, they dominated the Derby for a good while. The 'Dogs of War' era for them was then they last had success. Roy Evans and his squad just couldn't handle their aggression. I think it's this success with those tactics that make Evertonian's want the blood and thunder approach now.
As a Red I know that period in the mid to late 90's was a nightmare. Seeing them beat or drew against us all the time, and out battle us on the pitch was hard to take. Evertonian's got used to coming out on top on Derby days. At the same time they had very little else to shout about. It became their escape from relegation battles and bottom half finishes.
Once we got Houllier in charge and we were more physical, but also technically better, we started to dominate them. That was carried on by Rafa. We basically stripped away the one big thing that they drew some joy from season on season. I think that hurt Everton fans even though as a team they improved massively under Moyes. That's where the atmosphere around derby games really changed for me. I also think at this time Moyes played up to this with the 'People' Club' bit. Again it gave them something to hold onto, something to have a dig at us about.
Under Moyes they had their best period since the late 80's. He improved them massively. They just got beat by us a lot and we seemed to out do them even in their best moments. They come 4th, which was a brilliant achievement with the squad we had. We win the Champion's League. It's no wonder they don't like us and why the hatred for us has festered since the late 90's until now. There's Everton fans in their 20's and 30's that just really remember us beating them. That must be hard for them to take and it's resulted in the bile that some of them throw out there.
Where I disagree is about the importance of the game. For me, if I could choose any game to win in a season it would be Everton away. Followed by Everton at home. They aren't our rival for any trophy but on paper it's still the biggest game for me. Followed by the United game after that. No other club comes close to those 2. I find it mad that people get riled or upset about Chelsea, City, Arsenal or Spurs. I couldn't care less about those teams. In terms of long term rivalries at least. They are challengers for the trophies we want to win, in the same way that Real Madrid, Barcelona and Bayern Munich are.
I know it's easy to discount Everton as genuine rivals given our dominance over them but they still are our biggest rivals. Wait to see how much it hurts if/when they win a trophy again or they even manage to beat us at Anfield.