Let me try starting from the beginning and see if it helps... Sometimes I'm not good at finding the best wording.
A competition bow is usually placed on a high ponytail or pigtails secured with a plain, heavy-duty pony-o. The bow is then secured around the existing/base elastic- the one that is already holding the hair in place. The bow is not holding the ponytail/pigtail in place, and the purpose of the ponytail or the elastic attached to the bow isnt to secure the hair- only to secure the bow- which is a big deal during tumbling and stunting. It's got to stay, and it's got to stay exactly in place, and heaven help the mother whose child's bow so much as moves.
It's perfectly fine to use pony-o's on the competition bow, but flat elastic often provides better grip during tumbling and it has the added benefit of easier placement around the existing/ base pony-o. It can be both knotted repeatedl, and pulled a lot tighter than is possible with a pony-o. Here's how it works.
When you use a bow built on flat elastic, the ends are wrapped in opposite directions around the base pony-o and then knotted behind the ponytail. They can be knotted at each time the crossover in the back for extra security. The ends then pretty much just disapper behind the bow/ponytail- by which I mean you can't see them from a judging angle. Plus it doesn't require any additional securing from bobby pins. It's sturdy. To remove, you just untie the knots.
My daughter is a competitive cheerleader and we have our best luck with bows built on flat elastic, and the other mothers seem to prefer it as well. It's a nice option to offer, especially to competition or power tumbling squads.
Hope that makes it clearer and not worse.