You had to quit to get a raise
Suddenly you became more valuable after you give notice? It should make you wonder why you weren't valuable enough to deserve a raise before–when you were coming into the office every day and dutifully attending to your job duties.
Things won't change
The frustration, the stifling feelings, and the dissatisfaction that led you to seek new job opportunities will remain, and it's unlikely that the bump in pay will make those things any more bearable. Whatever turned you off about your job prior to the new offer will continue to be irksome after you accept it.
You walk into work, hand in your resignation to your boss, happy in the knowledge that you are going to a company with a better culture where you will be valued....and then your boss makes you a counter offer! What do you do?
You may be shunned
When you give notice, you are effectively, dumping your boss. As in many types of relationships, the snubbed party begins to bargain: Give me another chance....things will get better.... I can change! No one, after all, wants to be dumped! But once your boss' anxiety is eased and you've agreed to the counteroffer, new emotions will set in: resentment, suspicion, distrust. You will likely spend your remaining time at the company on the fringes–excised from the inner circle for your show of disloyalty (and co-workers may resent the raise and how you got it).
You're going to leave anyway
Four out of five employees who accept counteroffers end up leaving the company within six months, due to the reasons mentioned in the above four points.
You've already accepted an offer
And what about the new job offer you already accepted? By virtue of hiring you, that employer already has demonstrated a belief that you are valuable–and you haven't even had your first day yet. Your current employer, on the other hand, has begrudgingly offered you more money to get you to stay to suit his purposes. Also, leading on prospective employer–attending interviews, negotiating, accepting an offer, allowing them to think the job has been filled–is a bad career strategy in general.