Greatness can be distinguished from genius - although greatness is even less common than genius (and greatness may be - apparently - absent from areas of life where genius predominates - e.g. the theatre).
Genius is about ability - about outcomes, but although greatness requires abilities it is more about character - wisdom, maturity, dominance, solidity.
In English literature the premier genius is Shakespeare, and there are others such as Milton and Wordsworth - but none of these were great men.
The greatest figure in English literature was Samuel Johnson - who, although extremely able, was not a creative genius.
The contrast can be seen also between CS Lewis and JRR Tolkien - Lewis was a great man, while Tolkien was a genius.
It is possible to be both - Einstein was both a genius and a great man. Perhaps Rutherford too?
But you could be 'just' a genius - like most big names of the past: Newton, Darwin and so on.
I can't think for sure of any great men in science that were not also geniuses - maybe Lord Kelvin or David Hilbert? But maybe they too were geniuses?
Indeed, it is possible - maybe even usual - to be a rather small man and a genius - immaturity, foolishness, inconsistency, even insignificance are all compatible with genius. That was the contention of the play and movie Amadeus, and there are plenty of other examples. As an extreme and recent example, although I would not rate Picasso as a genius, most people do - and he was a very small man, qua man (and not just in stature).
Greatness is generally associate with politics and generalship - perhaps because effective p&g especially benefit from the same constellation of aptitudes as are found in greatness - however greatness is still rare.
I have already named King Alfred the Great as the prime example in English history; it is hard to find another so clear cut.
Perhaps there were great English generals (I'm not sure of this) - maybe Wellington, maybe Cromwell?
Perhaps the explorer Captain James Cook?
An example of greatness from my part of the world might be the railway engineer George Stephenson - he seems to have struck contemporaries in that way.
In US history the greatest man of whom I am aware is Robert E Lee.
Another possible 'great American' was William James.
Robert Frost (the poet - himself the premier genius of literature that America has produced, and who took his middle name from Robert E Lee) named the three greatest Americans as Washington, Lincoln and Ralph Waldo Emerson. I'm not sure about W and L but Emerson was a genius rather than a great man - which illustrates the difficulty of this distinction.
The greatest Scottish writer was undoubtedly Sir Walter Scott - despite that many would now deny him the title of genius.
Maybe King David the First of Scotland was their nearest equivalent to Albert (although, naturally, not so great!)
Greatness in music is of course rare - and seems likely to be found more among conductors and performers than composers.
For recent examples, the conductor Otto Klemperer and singer Dietrich Fischer-Dieskau seem to have had aspects of greatness - although the activities of being a conductor and singer are not themselves first rate creative activities in the way that composition is first rate - and despite that DFD was not even a first rate singer qua singer, in my opinion (lacking in tonal beauty and musical spontaneity).
Among painters only Rembrandt strikes me as great; among sculptors only Rodin. Although both of these were also geniuses - so do not make perfect examples.