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Math uses "Σ", as does modern Greek. But according to Wikipedia, "Σ" disappeared during late antiquity and the Middle Ages:
In handwritten Greek during the Hellenistic period (4th–3rd century BC), the epigraphic form of Σ was simplified into a C-like shape,[5] which has also been found on coins from the 4th century BC onward.[6] This became the universal standard form of sigma during late antiquity and the Middle Ages.
So when and why did it make a comeback?
Still, the lower case σ,ς are more lunate than not...
In this blog at the British Library you may monitor the transition from manuscript to print (Aldus Manutius) in late 15th century. The Florentine Homer manuscript (1466) of the British library still uses C.
In the first Aldine Greek book in print (1495),¹ however, you see Σ,σ, and final ς (an evident lunate holdover) supplanting their lunate analogs you'd see in contemporary manuscripts; lower-case σ, a less evident lunate morph, features routinely in byzantine manuscripts as well, e.g. 11th century, even 9th century; a Greek minuscule development expert might be able to track its history.
I am no expert to know the reasons Aldus Manutius had the print font Σ "restore" the non-lunate form, but it looks like once he got the ball rolling, the print world never looked back. This is the system the West inherited; you might be amused by Newton's mid-17th century notebooks.
¹ It appears to replicate an earlier, first printed book entirely in Greek, "Erotemata or Grammatica" of Constantine Lascaris (1434–1501), produced in Milan in 1476, in the printing house of Diogini da Paravicino (Dionysius Parvisinus). It still uses the lunate C!
This comes down to the distinction between "uppercase" and "lowercase" letters, which developed out of an earlier (by which I mean Mediaeval, not Classical) manuscript tradition that used special letters to emphasize certain names and the start of a new section. With the advent of the printing press, this distinction became standard in most typeset works; in Cosmas's answer you can see that it appeared in the very first book printed in Greek.
These special letters generally imitated the forms used in ancient inscriptions (sometimes variously embellished or illuminated), while the rest of the letters were based on the handwriting traditions that had developed in the intervening centuries. As a result, "uppercase" sigma looks like an ancient inscriptional Σ, while "lowercase" sigma looks like a handwritten σ or ς or ϲ.
(Most of this development actually happened in the Latin alphabet, but since the Greek alphabet had the same clear distinction between the ancient inscriptional forms and the modern handwritten forms, the custom transferred over easily when the printing press was adapted to Greek. The reason Cyrillic uppercase and lowercase letters tend to look the same is because they borrowed this tradition, but didn't have a distinction between ancient inscriptions and modern handwriting to draw on.)
