Eugene Wigner on von Neumann:
I have known a great many intelligent people in my life. I knew Max Planck, Max von Laue, and Wemer Heisenberg. Paul Dirac was my brother-in-Iaw; Leo Szilard and Edward Teller have been among my closest friends; and Albert Einstein was a good friend, too. And I have known many of the brightest younger scientists. But none of them had a mind as quick and acute as Jancsi von Neumann. I have often remarked this in the presence of those men, and no one ever disputed me. You saw immediately the quickness and power of von Neumann's mind. He understood mathematical problems not only in their initial aspect, but in their full complexity. Swiftly, effortlessly, he delved deeply into the details of the most complex scientific problem. He retained it all. His mind seemed a perfect instrument, with gears machined to mesh accurately to one thousandth of an inch.
Eugene Wigner on Einstein:
Einstein was a world-famous genius and people I knew used to remark, "You spend a good deal of time with Einstein. He has a perfect brain, doesn't he?" Well, I have never known what is meant by a "perfect brain." I do know that Einstein's mind was very human and had many defects. Einstein was far slower than Jancsi von Neumann to derive mathematical identities. His memory could be faulty, at least after 1933. And he was hardly interested in the details of physics. For a man like Edward Teller, developing the details of a physics problem was passionately important. For Einstein, it was not. In all spheres of life, Einstein's greatest pleasure was in finding, and later expressing, basic principles. But Einstein's understanding was deeper than even Jancsi von Neumann's. His mind was both more penetrating and more original than von Neumann's. And that is a very remarkable statement. Einstein took an extraordinary pleasure in invention. Two of his greatest inventions are the Special and General Theories of Relativity; and for all of Jancsi's brilliance, he never produced anything so original. No modern physicist has.
Jonathan Who comparing Einstein and von Neumann:
It's an interesting question to me in comparing the two. They are both regarded as geniuses but of each a completely different calibre. Einstein is what I'd call the "traditional" genius, he was a deep, contemplative thinker. He spent lots of time thinking through his ideas and he liked as Wigner said expressing simple ideas and basic principles. Von Neumann was the opposite, he was a fast logical thinker who would either solve problems instantly (ok technically I mean within a day or two) or move onto some other problem. He didn't seem to care much about the beauty or simplicity of an equation or idea either, for him it was interesting only to the point of solving the problem.
Jonathan Who comparing Grothendieck and von Neumann:
Grothendieck encapsulated the overall direction of pure mathematics towards abstraction and algebraisation in the second half of the 20th century (which continues to this day), particularly in algebraic geometry, algebraic topology and algebraic number theory, von Neumann was perhaps the opposite, he started in purely abstract fields (set theory and proof theory) and worked his way towards applied topics, eventually becoming arguably the top applied mathematician of the 20th century. Grothendieck died in seclusion at old age after giving up research mathematics for moral reasons while von Neumann died young still advising the US government till the end.
Michael Atiyah has said he calls only three people geniuses: Wolfgang Mozart, Srinivasa Ramanujan, and Johnny von Neumann.
John von Neumann died of cancer at age 53. Lewis Strauss (Chairman of the AEC) attests to Von Neumann's importance even on his death bed:
Until the last, he continued to be a member of the Commission and chairman of an important advisory committee to the Defense Department. On one dramatic occasion, I was present at a meeting at Walter Reed Hospital, where, gathered around Johnny's bedside were the Secretary of Defense and his deputies, the Secretaries of the three Armed Services, and all the military Chiefs of Staff. The central figure was a young man who but a few years before had come to the United States as an immigrant.