How can I overcome this nervousness at a horse show?

Don't compare your weaknesses and strengths with their's - instead, focus on your strengths and imagine yourself working the pattern flawlessly. Take deep breaths and focus on your horse, not the people around you. As soon as you start thinking negatively your horse will pick up on that and you'll usually end up blowing something. Always think positively, keep a smile on your face, and just have fun!