I guess if you take the entire universe of fragrances, these two would swim in the same lane. But just taking each for what it is and comparing them, they're not alike. I do prefer the Elixir over BDC Parfum.
Bleu Parfum is a woodier (read amberwoods) play on the original BdC theme, whereas Dior's offering is a much woodier and smokier theme than the original Sauvage. The Dior is also much stronger to the point that one spray gave me a headache.
I wonder if the woody aromachemical is the culprit, rather than the concentration. I notice it felt like an ice pick digging into my brain. I've detected this same aromachemical in Explorer and other Aventus copycats.This aromachemical is trendy right now. Even Abercrombie & Fitch is using it for Authentic Man.
"Which one is better?" - I'd say Sauvage Elixir. Bleu Parfum smells good but the base is a bit too sweet, and it's nearly imperceptible on my skin after a couple of hours. Sauvage Elixir just feels better crafted. Although it's pricier.