The best quote I can find which demonstrates the difference between the two words is :
1866 W. R. Alger Solitudes Nat. & Man iii. 120 Aspiration is a pure upward desire for excellence, without side-references; ambition is an inflamed desire to surpass others.. . . OED-3
The above quote is from the OED entry for 'ambition' which defines it as :
The ardent (in early usage, inordinate) desire to rise to high position, or to attain rank, influence, distinction or other preferment.
Whereas the OED makes clear that 'aspiration' originally comes from the the concept of breathing, aspiring and so an 'aspiration' is not a matter of striving for a place above others or to gain more than one has, it is the breathings of a human after something with purpose, for its own sake :
The action of aspiring; steadfast desire or longing for something above one.. . OED-3
1866 W. R. Alger Solitudes Nature & Man iii. 120 Aspiration is a pure upward desire for excellence.
I have an ambition to gain reputation points on EL&U.
But I also have an aspiration to be genuinely competent in English, that I may, effectively, communicate with my fellow humans.
There is nothing wrong with either as long as they are in harmony, I would say.
If not, I become an unbalanced human being.