There will be spoilers here. Of course.

According to my readings, in the book as well as the play that was adapted from it, Christine does indeed die from shooting herself in the head. However, the sound of the gunshot alerts the landlady, who reaches Rhoda in time to save her. Nobody else ever finds out what Rhoda is, because Christine had destroyed the evidence. Rhoda is therefore free to keep on killing, and fooling everybody.

When the first movie was made, Hollywood filmmakers were censored by the Hays Code. The Bad Seed had to have its ending changed because you couldn't show people profiting by (getting away with) crime. Therefore, Christine is allowed to live but the evil child had to die.

I don't know what to expect from the remake. I might see it out of sheer curiosity, but I can tell you this right now. Just one glimpse of some stills of the new "Rhoda" show me nothing more than a little girl who has been told to sit/stand on her mark with a blank expression on her face. Big deal. To this day, however, one still or one tiny clip of Patty McCormack in that role and I get shivers right down to the bone marrow.

Not to mention the performances from the rest of the cast, especially Eileen Heckart. You just can't fill those shoes.