Notre Dame is a very selective school, though obviously not as much as any of the Ivys. The thing is that it's just a way different experience. Notre Dame is Notre Dame... and if you've ever been there on a football game day, you'll know that no Ivy can touch that atmosphere.
Clearly it is an extremely strong academic school, though not as strong as the Ivys. Connections coming from either are strong, and ND can (depending on the reviewer and possibly football preferences

) carry as much sway as an Ivy education... regardless, Notre Dame on a resume will still draw a lot of attention. There is no lack of rigor, and religion is clearly a factor, but it is absolutely not a requirement.
Honestly, it comes down to what you're looking for. I'm all but narrowed down to ND and UMich.. two schools with amazing academics, school spirit and opportunites... those factors are a huge part of my preference.. I really love ND, so I may have some bias but I think I've told you straight-forward the way things are.