Let's consider the issue of parallel transport in relation to the figure on the following Wikipedia link: http://en.wikipedia.org/wiki/Parallel_transport
With reference to the Figure on the link:
Instead of parallel transporting the vector from A to N lets (parallel) transport it from A to N' along the meridian where N' is a point just below N [say latitude=89.9999 degrees]. Now we move the vector parallel to itself along the line of latitude passing through N'to reach the corresponding point on the meridian NB.The vector is now almost parallel to the meridian NB[Since the concerned line of latitude is not a geodesic I have used the word almost]. The vector is moved down and then moved back to A along the equator. It turns by a very small amount.The exclusion of a tiny[you could make it microscopic] spherical triangle is causing so much of difference.Why?