What is the difference between Minkowski Space and Flat Metric? A metric with a zero Riemann tensor is a flat metric, as is the Minkoswki metric but these metrics can clearly be different. What is the difference between a Minkowski metric and one with zero curvature but non-zero Christoffel symbols and non-zero derivatives of the Christoffel symbols?
 A: Flat spacetime has, as you say, a zero Riemann tensor and the Riemann tensor will be zero in all coordinate systems. However the metric will look different in different coordinate systems. For example the Minkowski metric is:
$$ ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 $$
while the Rindler metric is:
$$ ds^2 = -\left(1 + \frac{a}{c^2}x \right)^2 c^2 dt^2 + dx^2 + dy^2 + dz^2 $$
They both describe the same spacetime and they look different only because different coordinates are being used. If you compute the Riemann tensor from the two metrics both will give zero. If you calculate the Christoffel symbols from the two metric you will find they are different, but the Christoffel symbols are not tensors and they depend on the coordinate system being used. The difference in the Christoffel symbols is just a consequence of the difference in the coordinates.
We give the metrics different names only for historical reasons - Minkowski wrote down the first one and Rindler wrote down the second one.
