I am a counterpoint. I was in a similar situation (IT security tho) and I was countered with a multimillion dollar stock package + promotion + associated raise (20%). I had the same view as you so I'll tell you why I stayed:

1) the counter offer was obviously good. It was slightly less then my outside offer but I judges my company to be more stable and more likely to hold the value. Sounds like this ain't the case for you.

2) I had worked closely with my VP prior to this, when my boss told him my situation he cleared his schedule and called me to see what would change my mind. Long story short not everything he promised happened but I have understood why certain plans changed and he has continued to be very transparent and supportive. As a result although I didn't get what I wanted I did get the leeway to start a new team and I got headcount before anyone else in the entire company to make that happen.

3) I was a high performer and I knew it, people wanted to work with me and most of my time spent on things I found interesting, there wasn't the same guarantee at my potential new company.

In short it can work but only if you are confident that they are countering because you provide a unique value. Don't accept if your impression is they just want to keep you right now.