There are 2 things going on here. First of all, your little girl is still a little girl. It is difficult to remember this when she is highly intelligent. It shows up because the other kids have learned to hold in these natural tendencies to be goofy and silly. Your daughter is more of a free spirit or does not know what the others have already learned. To hold all of that in while at school and in public.
When you think about it 8 is so young on this earth. She has been around others that are serious and probably really too busy to just play with her as much as she needs. The other thing is that emotionally she is probably just immature and does not have many social skills. These skills are just as important as reading and math. She needs to be taught the social queues that we all have learned to help "read" what others are thinking, or feeling. She has not learned this.
The best way is to teach her this is through the rules of our society and manners. Once she learns these, she will better understand what is expected in all types of situations.
The other thing is she needs to be free to have fun, play, be silly, goofy and creative. It is hard to hold it together all day. Imagine holding all of this natural energy in all day and then having to go to clubs, music lessons or other activities and then going home and doing homework? At 8 years old?
The other thing is that if she is surrounded by competition and stress all day, she needs a way to vent. My child has never been into competition. Especially at school. We and the schools we sent her to all just wanted each child to do THEIR best. It was nobody elses business to know what others grades were. Same with parents. Instead we cheered any and all achievement and encouraged and helped with any and all struggles. We even taught our children it was ok and natural to make mistakes or to not always win or be the best.
I purchased a great book while my child was in elementary school it is is called "How Rude" The Teenagers Guide to Good manners, Proper Behavior and Not Grossing people out. It is funny and is still age appropriate for both your son and your daughter. I think it will help to remind her what the expectations are.
Sounds also like she needs some down time. Let her have some time off to just play. Each of you in the family need to spend time with her every week doing fun things. She is begging for attention.