After we learn basic Riemannian geometry, we see that there are several notions of curvature that are useful. When studying curves, we have measures of bending (curvature) and measures of twisting (torsion), but why does torsion not get the same treatment as curvature does in Riemannian geometry? Is this measure of twisting not as useful, or perhaps tougher computationally, as the measure of bending?
Here is the practical answer from a physicist: In the torsion-free case $T=0$, there can still be a non-trivial curvature going on. In particular the Levi-Civita-connection is torsion free, which is the only connection needed for General Relativity. (well, you can do GR with torsion, but that is purely hypothetical, and not very useful afaik).
The answer is that in Riemannian geometry one studies Riemannian metrics. Riemannian connections come as a useful tool for this study. Each Riemannian metric has many compatible connections (connections such that the parallel transport preserves the metric: that is, unless the manifold is 1-dimensional in which case there is a unique compatible connection). Among these connections, there is one which is easiest to work with, namely, the unique torsion-free connection (the Levi-Civita connection). Using it over other connections makes calculations much simpler. See also the discussion at Mathoverflow on the intuitive meaning of torsion for connections.
Another reason to like Levi-Civita connection is that pull-back of the flat connection on $R^n$ to an isometrically embedded submanifold is again Levi-Civita, which makes it a natural concept.
As I said in my comment, there is no relation between torsion of curves and torsion of connections, apart from the name. I am not sure who is responsible for this terminological confusion.