Oh boy. I can relate. I have a very similar dynamic with my two kids (my gymmie DD 12 and DS 9). I’ll share a few things that have been helpful for me.
First was the realization that my kids are uber competitive. Competition drives everything for them and they both want ‘best kid’ status when it comes to me or DH. The competition factor alone still doesn’t explain why my DD is so much more vicious in her comments toward DS. My DD has the added bonus of being an introvert and she also has anxiety disorder. Her lash outs against her brother follow a pattern where she is trying to build herself back up or protect her self-esteem or she is truly annoyed with a more extroverted way of doing something.
Second, the book Siblings without Rivalry was very helpful to me. It was here that I realized I needed to change how I talk about the sibling that isn’t around. For example, in my head I think “I can’t wait for DD to come home. It’s been 14 hours since I’ve seen her and I miss her! I can’t wait to hear about her day and oh yeah, that giant…I wonder if she got it today.” Before I read the book I would tell DS this, “DD will be home soon. I have to make her dinner. We will see if she got her giant today.” What I now tell DS is that “I’m making dinner.” Of course I have the same excitement about seeing DS at the end of his day but he lacks perspective to see this – instead he was hearing that I was taking special interest in his sister’s accomplishment. Although I actively minimize what I say about them it has not minimized our family’s ability to share in each other’s successes. DS was already aware that DD was working on a major gym milestone. I didn’t need to expand on this and really it’s DD’s story to tell anyways. Since I’ve backed off talking about them they ask each other for more details about their day.
Now back to DD and the self-esteem/anxiety. She struggles with self-esteem and especially how others perceive her. Her inner thought process has her thinking that if she laughs at a joke her brother tells then she acknowledges that he is witty (and the implication is that she is not). If I compliment DS on something related to schoolwork she trys to stump him with a difficult question to ‘prove’ that he is not smart (because if he is smart then people may think she isn’t smart). It goes on and on. (She’s not like this with friends and teammates, she is able to freely offer support and compliments even though she admits it sometimes makes her feel bad about herself). She was in therapy for anxiety a few years ago so we have the vocabulary to talk about much of this. I’ve also laid out my expectations of how she is to talk with her brother (little stuff, like she will tell him good bye in the morning and good night before bed). One time DS got an award at school. I knew DD would try to belittle his accomplishment. Before he told his news I gave her a heads up that he has news to share and what my expectations were for her reaction (she did fine with him and the two even engaged in a conversation about it). At first it was very forced and so obvious that DD was pained to be saying something positive to her brother. Slowly though it’s changed and she is in a much better position now to offer genuine praise and compliments.
If any of what I shared rings true, I suggest the book. I've never made overt comparisons to their face and I feel I always respected their individuality. But I’m 100% guilty of subtle comments that could easily be interpreted as comparisons by my kids who also happen to be very competitive individuals. I have no clue if any of these things have helped my kids get along but since things are much improved now I’m running with it! My kids still bicker but they can have real conversations now and DD isn't always belittling or minimizing her brother.
D1 is the reader in our family (completed the Harry Potter series by age 10)...and tested into a G&T program during the transition from elementary to middle school. It was night & day for her once she was no longer "waiting" for her classmates (more than once, she was "disruptive" in class because she had already completed the work and had nothing to do). One of my key learnings from her time in the G&T program was the concept that Talent Development was not pre-paced, but rather at the learner's own (typically accelerated) pace (we were also told not to expect "balance" with these kids). And yes, I sometimes wonder how that applies to D2 and gym, but let's not go there for now. At any rate, her growth during this time was phenomenal. Though personal and academic growth rates may have differed, the academic stretch definitely helped the personal. The key point here for children with gifts, is to really let them run with it once they find their niche or strength. Do everything you can to nurture it, support it, enhance it. 
