Part of Elaida's problem is that she seemed to have spent very little time out in a hostile world. During her training the impression I got was that she was the star pupil because she learned so fast and everyone knew that she was going to be among the strongest of her generation.

Then she graduates and spends time in the Tower that is not exactly hostile to Aes Sedai. She may have been invovled in the vileness but if so she was part of a team. Then she goes to Andor and the royal court that is about as pro Aes Sedai as its possible to be outside of the Borderlands.

Elaida never had to go out into the world in places like Illian or Chairhien or the west coast that were not rabid anti Aes Sedai but equally did not bow and scrape. Elaida never had to face a situation where she had to negotiate or work with ordinary people or nobles. Where she went she was shown deference and respect if not downright obedience.

Elayne grew up in this same environment but she did not have Elaida's inheritent character flaws and was taught by her mother how to play the game to get what you want when you can't simply give orders.

When she gets real power she had never learned to respect the opinions of anyone who is not Aes Sedai and worse I suspect she simply does not understand people so she can't relate to them or get them on side.