Differences in how mathematical results are proved in the time of Euclid and in the twentieth century What is the difference in the manner demonstrated in Euclidean time and as demonstrated in the twentieth century?

A very very big one indeed ! 
There a big differences in style : today math is highly symbolic, Greek math is not so, and in content : we thik to math a science of "strucutres", for ancient Greeks geometry is "the science of space", arithmetic is "the science of numbers", (geometrical) optics is "the science of light rays". 
But there was a great transformation from ancient Greek mathematics to Euclid and Archimedes (whith which we are more ... accustomed). Ancient Pythagorean "numerology" is very far from modern number theory, while Euclid's or Archimedes' poofs are "quite easy" to understand.
You have to read at least : 


*

*Arpad Szabò, The Beginnings of Greek Mathematics (1978),

*Wilbur Richard Knorr, The Ancient Tradition of Geometric Problems (1st ed 1986 -
Dover reprint), 

*Wilbur Richard Knorr, The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry (1975), 

*Ian Mueller, Philosophy of Mathematics and Deductive Structure in Euclid's Elements (1st ed 1981 - Dover reprint),

*Reviel Netz, The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History (2003) 
and a good history of Greek mathematics, like :


*

*Sir Thomas Heath, A History of Greek Mathematics (1st ed 1921 - Dover reprint, 2 vols). 


But, despite this big difference, we are able to "read" and understand an ancient proof (e.g.Euclid's Elements) without too much difficulties; and this fact is characteristic of mathematics that ancient science do not necessarily share.
Aristotle's physics and biology are only interesting for historian of science, while Euclid's Elements is ... mathematics.
