Thanks all for the replies. In this case, I was a fairly key performer but I believe it is that company just doesn't want to pay any more. And they are happy replacing me with somebody else
It's worth considering the idea that the question is not, "Why did my old company value me so low," but, "Why does the new company value me so highly?"
It could be a million reasons. They're flush with cash and gearing up for a big expansion. They lost someone key and will pay extra for a quick hire. They're actually paying about the same because their benefits are awful (or more optimistically, their benefits are equivalent but better-negotiated). Someone there knows you and sang your praises. The hiring manager thinks you're sexy. They're incompetent and don't realize they're overpaying. Who knows?
Bottom line is that showing an offer to your current employer is always a gamble. Personally I think you're better off; the work atmosphere after a stunt like that can be unpleasant if your boss feels like she was strong-armed.
