Consider why are they willing to raise your salary when you were not expecting a raise for some time? The reason is that when a resignation is tendered an employer can often obtain a quick fix by throwing money at the problem. Recruiters, employment advertising, training costs all affect a department's budget. Why spend that kind of money when some well applied pressure might turn you around and solve the problem? It is much cheaper to keep you - even at a higher salary.
Employer-managers are concerned that they may look bad, and this could affect their standing because they are judged by their superiors partly by their ability to retain staff. When a contributor quits, department morale may be affected. Further, your leaving might jeopardize an important project, cause a greater workload, or affect the holiday schedule. It's never a good time for someone to quit, and it may prove very time consuming to replace you. Some employers will actually tell you that your counter-offer is usually a stopgap measure because they couldn't afford a defection at that point in time. The pressure of having to make a counter-offer can often affect the level of future trust between the hiring manager and the employee.
Your interests will be secondary to your company's interests. While your employer may truly consider you an asset, and may genuinely care about you as a human being, you can be sure that your interests are secondary to your company's. In other words, tempting offers and comments are attempts to convince you to do something that is in your employer's best interests, and not necessarily yours. Where did the money or new responsibility come from? Was it your next raise just early? Will you be limited in salary growth in the future? Will you have to threaten to quit to get your next raise? If the department is so dependent on one person leaving, then the company has more fundamental operational problems and issues.
You'll likely not be considered a loyal team player again. You've demonstrated your unhappiness, or your lack of team loyalty. Many employers will remember this at the next review period, and trust, once broken, is very difficult to re-establish. Statistics show that the majority of those accepting counter-offers leave, or are terminated, within 6-18 months. Apart from a short-term solution and treatment, nothing really changes. The reasons why you started looking are still there. After the dust settles, the same problems very often reappear.
A counter-offer demonstrates disrespect for your decision and commitment to the new company. A decision to make a career move is a personal choice not to be taken lightly. When you submit your resignation, it should be after having weighed carefully the pros and cons of making the move. You've committed to the new company, which has made plans and preparations for you. They are counting on you to act responsibly. Don't sell out, or back out. Stand by your word. Everyone will respect your integrity. Don't let anyone tell you directly or indirectly that you shouldn't trust your own intuition and logic. After all these are your dreams and aspirations!