Sure there could be an advantage... if your 'home' page was geared to steer (or funnel) traffic towards one part of the site, you might make it different from the 'index' page. I can't think of a specific example, but I suppose someone might want to...
This is just a question of internal links... you could replace 'index' with any topic-specific page, and just never link to your 'home' page from within the site. Maybe that's a better way of approaching this question. However, I doubt there's any advantage for Organic
SEO... it's probably best used as a traffic funnel.