The major difference between New Kingdom Egypt and the older systems was that the New Kingdom was more modernized and professionalized and was also more aggressive in its foreign policy.
In the New Kingdom, the military became a much more professional organization. This was in contrast to the old system in which regional governors would get up what men they could and come to fight for the Pharaoh when needed. As the military became more professionalized, it also became more powerful in political terms. In the New Kingdom, viziers (the most important bureaucrats, people who actually ran the kingdom) came only from the military.
With this more powerful military (and the new weapons and tactics the Egyptians had learned from the Hyksos), Egypt became more aggressive in its foreign policy. It was at this point that Egypt conquered Nubia and occupied and controlled areas of Palestine and Phoenicia.
In these ways, the New Kingdom was more organized and modern than the older kingdoms. This allowed Egypt to become more of a regional power with something of an empire.