From a Scouse Liverpool fan, the rivalry between us always used to be big brother to little brother kind of thing. I'm told in the 80s times were great as it was the two of us at the top most seasons.
Things have soured since and Everton spend a lot of time circling the drain. As such they have become quite bitter(our nickname for them was The Bitters for a long time). All they have now is "well at least Liverpool didn't win X." There is another fanbase going the same way but they're not fully there yet.
As such, even we're just kind of sick of them now.