python - Difference between Vincenty and great-circle distance calculations?

Python's geopy package features two distance measurements techniques: Great Circle and Vincenty's formulae.

>>> from geopy.distance import great_circle
>>> from geopy.distance import vincenty
>>> p1 = (31.8300167,35.0662833) # (lat, lon) - https://goo.gl/maps/TQwDd
>>> p2 = (31.8300000,35.0708167) # (lat, lon) - https://goo.gl/maps/lHrrg
>>> vincenty(p1, p2).meters
429.16765838976664

>>> great_circle(p3, p4).meters
428.4088367903001

What is the difference? Which distance measurement is preferred?

Answer

According to Wikipedia, Vincenty's formula is slower but more accurate:

Vincenty's formulae are two related iterative methods used in geodesy to calculate the distance between two points on the surface of a spheroid, developed by Thaddeus Vincenty (1975a) They are based on the assumption that the figure of the Earth is an oblate spheroid, and hence are more accurate than methods such as great-circle distance which assume a spherical Earth.

The accuracy difference is ~0.17% in a 428 meters distance in Israel. I've made a quick-and-dirty speed test:

       : Total 0:00:04.125913, (0:00:00.000041 per calculation)
   : Total 0:00:02.467479, (0:00:00.000024 per calculation)

Code:

import datetime
from geopy.distance import great_circle
from geopy.distance import vincenty
p1 = (31.8300167,35.0662833)
p2 = (31.83,35.0708167)


NUM_TESTS = 100000
for strategy in vincenty, great_circle:
    before = datetime.datetime.now()
    for i in range(NUM_TESTS):
        d=strategy(p1, p2).meters
    after = datetime.datetime.now()
    duration = after-before
    print "%-40s: Total %s, (%s per calculation)" % (strategy, duration, duration/NUM_TESTS)

To conclude: Vincenty's formula is doubles the calculation time compared to great-circle, and its accuracy gain at the point tested is ~0.17%.

Since the calculation time is negligible, Vincenty's formula is preferred for every practical need.

Update: Following the insightful comments by whuber and cffk's and cffk's answer, I agree that the accuracy gain should be compared with the error, not the measurement. Hence, Vincenty's formula is a few orders of magnitude more accurate, not ~0.17%.