Is there a way of differentiating between different sects of chassidim by the type and color of socks that they wear as well as by how they wear them?
You cannot determine sect solely by the socks, but often you can tell something based on the socks. Combining sock style with hat/jacket/girdle/shoe/glasses/shirt style will narrow things down even further, as will observation of an individual on both weekdays, shabbos, and holidays. Together with knowledge of which sects are common locally, it is often possible to determine exact sect with high accuracy.
All the following are generalizations; each sect has individuals which differ:
In many sects, only married men wear high black socks on weekdays.
In Belz and related sects, even unmarried men wear that.
Satmar and related sects often wear long pants on weekdays.
In many sects, married men wear high white socks on shabbos and holidays.
In Satmar, many wear long pants even on shabbos and holidays.
In Belz and related sects, married men wear high black socks on shabbos, but white on holidays.
In some Jerusalem-based chassidic sects (especially for sect names that begin with Toldos) even unmarried men wear high white socks on shabbos and holidays.
If someone is wearing high white socks on a weekday without wearing a shtreimel (because that would indicate a personal festive day) you can safely assume them to be of a rabbinic family.
Ger and related groups wear high black socks with normal pants which are tucked into the socks – all week.
Other "Russian" chassidic sects (e.g. Chabad and Karlin) never wear high socks.
Source: Living in chassidic neigborhoods for over a decade.
I wear high black socks on weekdays and white ones on shabbos and holidays.