Usually, you don't use a job offer at another company as leverage to get a counter and a raise at your current employer.
If you accept a counter, they think you are all about money, and I can guarantee you they will be looking for a replacement (a cheaper one) and/or transfering your duties. Consider this a management rule.
You are only as good as your word, if you already took action to get a raise and compensated in oter ways at your current employer yet still found an outside offer more intriguing, then there shouldn't be anything that changes your mind. You never want to threaten anyone, and idle threats are 10X worse. So go with your word, nothing should disuade you right?
UNLESS - I'd say of the counter in querstion is 25% higher, its considerable, BUT, you will lose respect from some of your peers as you would then be in it just for the money, AND you would burn a bridge with your other prospect.
I'd stick with your word. If you made a decision to switch after weighing options and going through all the hassles, I think your mind is made up.
Ironically enough, next week is my last week at my current employer, and I'll be moving on as well. They did counter offer, and I said to them,
"I tried telling you about the situation, gave you subtle and direct hints, even directly asked for some other forms of compensation. Now, when another company recognizes what I have to offer, and says they will provide what you said you cannot, you want me to remain loyal? If I tender my resignation, consider it too late, I'm sorry. But please use this as a lesson learned for others who are still here."
Good luck.