It was t important for them Lizzie thought it was just a dress some guy had a crush on Caroline and josie was just saying to lizzie that the Dress was from klaus

Yeah but I mean why did they want to wear that specific dress for the competition I mean it had nothing to do with Caroline’s pageant

The argument wasn't about the dress, the argument was mainly about Lizzie's behavior. Josie tells Lizzie that by choosing Hope as a future winner she never took into account that her could hurt her feelings. Josie just adds that if Lizzie doesn't care about her sister's feelings she could at least take care of Hope's feelings. Of course Lizzie didn't know about the dress.

Besides, I don't have the memories that Josie or Lizzie wanted to wear this dress.

During the argument Josie says “do you know how good it would feel to compete in moms dress” or something like that like it’s something important and for them in this situation it’s not

Well, Caroline probably said that dress was really special to her and loved it. Wearing something that could be your mom favorite dress for pageant and win with it must feel special and an honnor, I don't know...