She needs activities all her own-- not ones that are copies of what her older sister does. She should get to choose those activities (but you need to direct her toward things that aren't just like sister's). Is she in Girl Scouts? She would get to be her own person there (but be sure it's not the same troop her sister was in a few years back!). Does she have any interest or budding talent in art, music, dance, a sport?
Please, never expect her sister to include her in play dates at home, and never expect little sister to include big sister when one of little sister's friends is over. It seems fair but it isn't. Do you ensure that she has her own solo play dates with her own friends her age, and is not ever an "add-on" to her sister's play dates? She may say she wants to be in on her sister's play dates, but if older sister has someone over, try hard to invite one of younger sister's own friends for the same time, and have separate activities lined up -- your younger child is still young enough that you might want to have some specific craft set up for her and her friend to do, to distract them away from the older girls. Or have kids over when your husband can oversee older sister's play date and you and younger child go out alone together. How much of your alone time does she get, or do you do everything with both of them together? Think about that. She may need more of your time.
Is she expected always or often to be at her sister's sports practices and/or games? If your older daughter is very involved iin sports -- well, I have seen families where the older kid's sports schedule gradually begins to dominate the whole family's schedule (and it gets worse as they get older). Before the family knows it, everything gets dropped if there is a game. Even if you and your husband have to split up the duties, ensure that little sister is not expected to attend every game, just the truly most significant ones, and during all other games and practices, ensure little sister is somewhere else doing HER activity.
Stop trying to reason with her. You can't. It's pure emotion she's operating on here, and reasoning will only make her defensive and more angry and unhappy. It sounds like, as another poster noted, she has learned she gets some attention if she is negative and unhappy, so she needs to unlearn that -- not by your scolding her, but by your ignoring it. (Scolding or punishing her for being unhappy only gives her "negative attention," and to a kid, that is still attention and still desirable, although adults don't get that.)
It really does sound like she's overshadowed and has found a role for herself in the family where she can get attention focused on her. Discourage that role with positive attention when her behavior is positive. Get her out from under her sister's shadow and take away the attention given for the negative behaviors. By the way, be sure that your own innermost feelings about her are not being damaged by this -- it would be easy even for a parent to prefer the bubbly, outgoing, smart sports star to the high-maintenance, hard-work, unhappy child. Parents don't mean or want to do this but it's human nature. And kids pick up on it.
One additional issue for this fall: If your children are in the same school, be SURE your younger one does not get the same teachers your older one had. Ask the school before the fall to ensure that this does not happen. Despite training to the contrary, teachers WILL compare one sibling to another and even mention it to the "problem child's" face. I've heard it and it's a real crusher for the younger, less bubbly sibling.
