Shortest possible answer: Pragmatism.

Shorter answer: The former (though later chronologically) threatened American interests and plans for the region while the latter was completely in line with them.

Short answer: Getting rid of Mosaddegh was meant to keep the U.S.S.R. and communist influence out, Western influence in control. The Suez Crisis was going to cause Soviet influence to rise around the world since the act was blatant colonial and Western aggression against newly independent state. The Iran coup kicked out Soviet influence (in the administration's mind), the invasion of Egypt "forced" Nasser to move closer to the Soviets, weakened every Middle Eastern leader who had ties with Britain or France, and put the Americans in a lose-lose situation.

I'm going to focus solely on the Suez Crisis here - I'm not in any way discounting the importance or fascinating topic which is the overthrow Mosaddegh, it's simply that the Suez Crisis in particular is an incredibly vivid example of the tensions in the Eisenhower administration's foreign policy making. I'm also going to argue that Eisenhower's response to the British was hardly severe, and in fact almost the bare minimum the U.S. could do - it was indisputably far less than what Nixon or Dulles wanted to do.

I want to preface all of this by flagging that I wrote my dissertation from a perspective which is, let's just say fairly critical, of Eisenhower and Dulles on topics like this, so everything I write should be understood as coming from a person who think Eisenhower often made very poor medium and long-term decisions.

One thing that needs to be understood right off the top is that Eisenhower's action against Britain was viewed by him and effectively everyone in the administration as the absolute minimum, (along with letting Nixon vent and attack the British whilst campaigning) that could be done to show the British and the world that the U.S. considered the British-French-Israeli attack to be inexcusable. Dulles and Nixon had wanted to cut the British off far more severely, but the prevailing opinion was of course Eisenhower's (and many in the State Department).

One reason for this reaction was that the invasion of the Suez effectively slammed, bolted, welded shut, then sealed off the door to any further pondering on Eisenhower's part as to which side the U.S. would throw its weight; Arab nationalists like Nasser, or Islamic monarchs like Saud. While the administration had typically swung more towards the Islamic leaders (by dint of the fact that religious types like Dulles and Eisenhower believed more stridently religious leaders were less likely to "fall" for godless communist offers or influence), there was this kind of "hard to get" air to Nasser that beguiled Dulles and Eisenhower. The State Department often came up with "fun" little plans to fix the area, like creating two federations in Northern Africa and in the Middle East out of states (Iraq would lead a federation made up of Jordan, Syria, and itself), which, unfortunately for the administration, were often dashed upon the rocks after local leaders either flatly rebuffed them, or, in Syria's case, literally joined Egypt to form a new state.

But, again, while Eisenhower's decisions result in the path the administration took, this shouldn't be taken as meaning it was the obvious or only path they thought they had. In fact, as the crisis was underway, something that to my knowledge occurred only once in the Eisenhower administration unfolded - the Secretary of State and President were at such opposite ends of the issue that John Foster Dulles became so disturbed and angry that he stormed out of the meeting.

To those familiar with Dulles, or rather, to those familiar with Dulles beyond the caricatured corporate demon he's often portrayed to be in more recent histories (thanks Kinzer), his loathing of British imperialism and colonialism will be well known. To those unfamiliar - Dulles once held up a meeting with British officials and demanded to see the treaty they referred to which they claimed supported their position; when the treaty was found and then presented to Dulles (the British had of course not brought a copy to the meeting, so the meeting was paused in order to find one), Dulles read it in front of the British officials, then flatly stated the treaty did not say anything like what the British claimed. That is, Dulles hated the British Empire. (He didn't hate the British, he hated empires, particularly stagnant, static ones).

So Dulles took the crisis as a golden opportunity, for the U.S. to begin to truly break away from the old empires. Dulles saw the invasion as a means for the U.S. to take a leadership role in the Middle East, by driving out the old empires. Dulles stated:

"For many years now the United States has been walking a tightrope between…our old and valued relations with our British and French allies…and…trying to assure ourselves of the friendship and understanding of the newly independent countries who have escaped from colonialism…[I]n view of the overwhelming Asian and African pressure upon us, we [can] not walk this tightrope much longer…In short, the United States would survive or go down on the basis of the fate of colonialism

(Discussion at the 302nd Meeting of the National Security Council, Thursday, November 1, 1956, 5, 302nd Meeting of the NSC, November 1 1956, Box 8, NSC Series, Ann Whitman File, Dwight D. Eisenhower Library).

Dulles was somewhat shocked, and grew evermore agitated that, not only did Eisenhower see things differently, the President considered Britain's actions in particular to be problematic but not unsurprising - contending that even if the British expanded the conflict in the area and brought in American supplied military aid to bolster their efforts, it would simply “represent another case of 'perfidious Albion’". This type of apologist attitude continued, and Dulles finally snapped. He interrupted Harold Stassen, and stated both to Stassen and the NSC, "We do not approve murder." The Secretary then got up, claimed he needed to put something on paper, and left the meeting.

(Fun fact: Dulles meant the murder thing literally. As in, "we don't say 'kill this person'." In fact, on 18 August, 1960, at the NSC meeting, Eisenhower turned to Allen Dulles and directly ordered him to have the CIA eliminate Congo's soon-to-be Prime Minister Patrice Lumumba. The room fell into a "stunned silence" that Eisenhower had said such a thing out loud. See Editorial Note, FRUS: 1964-1968, Volume XXIII Congo, 1960–1968, 19; Discussion at the 456th Meeting of the National Security Council, Thursday, August 18, 1960, 10, 456th Meeting of the NSC, August 18, 1960, Box 13, NSC Series, AWF, DDEL).

Really though, the steps taken against the British were not in any way, shape, or form designed to do anything other than sting and embarrass them. It did not end the U.K.'s reign as a global power. Britain is still a global power, it's just not a super power, but it wasn't one in 1956 either. And, Britain was still quite capable of flexing its muscles in the region after 1956; British troops helped save King Hussein's rule in Jordan in 1958, and in 1961 their presence in Kuwait made General Qasim of Iraq think better of trying his luck in annexing the region.

But back to 1956.

By 30 November of 1956, while the relationship between the Americans and the British and French had been shaken, Dulles described the acrimonious mood at the moment as “essentially a violent family squabble” that would pass with time. And, Nixon even pressed cabinet members on whether there was any means for the administration to aid Britain’s Conservative Party, as the Eisenhower administration was apparently terrified Nye Bevan’s rhetoric might place him back in a position of power. (Discussion at the 305th Meeting of the National Security Council, Thursday, November 30th, 1956, 10-11, 305th Meeting of the NSC, November 30th 1956, Box 8, NSC Series, AWF, DDEL).

Other than the archival references, I'll also toss in:

Sohail H. Hashmi. "‘Zero Plus Zero Plus Zero’: Pakistan, the Baghdad Pact, and the Suez Crisis," International History Review 33, No. 3 (September 2011). - Mostly because I think it's a good read.

Richard J. Alexander. "Couscous Mussolini: US perceptions of Gamal Abdel Nasser, the 1958 intervention in Lebanon and the origins of the US-Israeli special relationship." Cold War History Volume 11, Number 3 (August 2011): 363-385.