The only difference is that it is using Simplex method instead of gradient descent (as in article). Simplex and GD are slightly different methods of finding local minimums in systems of linear equations.
Cassowary is an incremental version of simplex—it lets you efficiently add, remove, and modify constraints.
Gradient descent works for non-linear systems, but isn't guaranteed to find the global optimum.
And within convex problems there is a hierarchy of "difficulty": LP ⊂ QP ⊂ SOCP ⊂ SDP
The Simplex algorithm solves only LP. And it was shown that the Simplex has an exponential worst-case runtime (in the number of constraints). Whereas interior-point methods (basically variations of Gradient Descent) have a much better worst-case runtime.
See here for further reference: http://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf