Egypt is central to the Arab world, along with Saudi Arabia.
It also controls the Suez Canal, vital for certain grave military contingencies, like a war with Iran in which one wanted to be able to move rapidly between the Black Sea and Persian Gulf.
Money from oil and Mecca-Madina make the Saudis a big dog in the Arab world, while demography and Suez makes Egypt its twin Arab Big Dog; there are a 83 million Egyptians.
The history of Egptian-American relations is a curions one, as Egyptians and Americans never quite seem on the same page, if it's a relationship that has hardly mainly been hostile.
As to benefits, Cairo's infrastructure was built orimarily by the United States, if that doesn't seem to have bought us a lot of goodwill lately. We have transferred a lot of food and more usefully agricultural technology to Egypt over the years, not a trivial consideration in feeding 83 million people.
As to our relationship with Egypt, like in much of the Middle East, it originated as part of the power transition from the British to the American Empire.
Until 1936, Egypt was a de facto province of the Empire, if a relatively lightly governed one, held by Britain because of the need to defend Suez.
Nationalist circles in Egypt still resented the level of British control after the disengagement agreements of 1936, which is where the Free Officers came from that were the base of Nasser's coup against the British sponsored monarchy in 1951, a coup we actually didn't resist at all.
America's role in that coup is an interesting one, if not talked about much. See Quicksand for details.
As to the motive, like later with King Idris in Libya, in the context of the Cold War, Egypt was considered the prize of the Middle East, in need of modernization to insulate it against Soviet influence, if the atheism of the Soviets was self-limiting. That is why Putin talks a lot about Orthodoxy, a return to Nicholas I statescraft, if that ended badly in Crimea.
Nasser seemed to be the answer to the lassitude of the late Farrouk years, as to like Qadhafi being a charsimatic and energetic young Colonel, leadng Egypt to the Promised Land of a modern society.
At the beginning of modern Egypt, Nasser was a friend at first of America, if one who proved to turn somewhat on his sponsor, probably seeking to both gain external space and to suppress the element we see now: the Muslim Brotherhood.
The Brotherhood emerged in Egypt in the twenties as the main popular alternative to Arab nationalism with a socialist bent a la Nasser, the Western bourgois-style liberalism of the Wafd never quite catching on for a reason vital to understand now; too close association with the West.
After around 1930, too close association with the West becomae the poison pill of Arab politics in general, if the Brotherhood types anywhere and everywhere scared native power elites as to being thought fundamentally unreasonable. Maybe there are, maybe they aren't, but a lot of people even within ruling Arab circles had grave doubts about ever allowing them to come to power, as to why it's not only because of America that Mubarak existed.
In any event, the Egyptian military that had reluctantly supported Britain in WWII, in fact at the point of the gun, also retained enough secularism to look the other way at best at the death of the founder of the Brotherhood.
People in Egypt who still respect Nasser seem to have forgotten that he executed chief Brotherhood theorist Sayyid Qutb as well in 1966, Qutb being a hero to jihaddists everywhere, including OBL. More on him in a minute as to a painful truth of life sometimes, that sometimes when you really get to know people, you like them less, not more, which is the long and short of Qutb and his Salafist ilk with respect to the United States, whatever we do save abandon Israel and convert to Islam.
In any event, as to the history of Egyptian-American relations, Nasser bit the hand that fed him hy accepting Soviet weapons in 1955, which led to a tit for tat exercise culminating in the Suex War, the last independent use of force by Britain in the Middle East.
We backed Nasser in that war against Britain, France, and Israel, in spite of him being something of a turncoat, something people in Egypt seem to not be very thankful about.
Similarly, after Nasser was replaced by Sadat, we prevented the Israelis from exterminating the Egyptian Third Army, as to a curious lack of gratitude. That's life.
Sadat of course is the source of the contemporary relationship as to the Peace Treaty with Israel and the associated aid package now it would seem coming under fire after the disturbances of the last week.
Sadat was always loved in the West more than in Egypt, partly because of the peace, and partly because of his move away from Nasser's bankrupt socialism.
It's not like America didn't put a lot of effort into building a modern Egypt, as to reasons to think carefully about radical course changes now.
Just four years ago, for all the criticism of Mubarak, Egypt was becoming something of a poster child for economic development. That's the Egypt we thought we knew in Tahrir Square in last year's protests, which for all the second guessing wasn't all wrong, even as it underestimated the influence of Salafists like Qutb of the Brotherhood, whose story is now worth telling.
Qutb lived in America in the 1940's, brought here as part of the same generosity towards Egypt that built the American University in Cairo.
The response of Qutb is alas not an untypical one of those who cross cultural boundaries, which was to come to passionately hate the United States, which he then propogated in various Islamist works, especially the role of women.
Those who would be ever critical of Mubarak' relationship with the United States ought to reflect on the fact that since to date the only people to emerge are people who idealize Qutb, we could have done much worse, just as those Egyptians who have always been our friends, including in the military, ought to see the if Mubarak was the bathwater of a decaying Nasserist system, the Brotherhood is treading on thin ice if it wants to be thought of as the baby, given the history of that relationship to date.
That is not an argument for antagonizing President Morsi, as to be fair to the Brothers, formally they have renounced the violence of Qutb and the jihaddists he inspired over the years, jihaddists who killed Sadat and jihaddists like Ayman al Zawahari, all former Brothers.
America has deep interests in Egypt if only because of Sinai, if also because of the peace with Israel, and also because of the intrinsic importance of Egypt as home to al Azhar and the largest population of Arabs.
One doesn't casually toss things like that aside, especially because even with Mubarak, given the constraints involved, including the obvious hostility of many Egyptians to America that has little to do with what we have done to them, since they hated the West in the twenties, we have in fact tried to do good there over the years, and have done some good, like feed a whole lot of people.