Siemens (along with our old friend collegeboard) runs the Siemens Competition. Intel runs two competitions, ISEF (International Science and Engineering Fair) and STS (Science Talent Search) which is only for seniors.
ISEF is like a traditional science fair with a board, judges, etc. Siemens is a write-in competition with a research report, abstract, the whole schabang. STS is an application process. I've done ISEF and Siemens, but not STS so don't know that much.
And yes you can submit the same project to multiple competitions. Just not same project same competition over the course of several years.