No, they are the same thing. What this sentence is referring to is the vanishing of the Jacobian determinant [1] (which is defined using the derivatives of the defining equations).

A simple example is the equations y^3 - x^2 = 0. This is a "cusp" (use wolfram alpha to see what it looks like) and has a singularity at the origin.

The jacobian is the matrix:

[ -2x, 3y^2 ]

This has rank 1 unless x and y are zero in which case it has rank zero. The fact that the rank is less than 1 indicates a singularity.

[1]: http://en.wikipedia.org/wiki/Singularity_(mathematics)#Algeb...