I fully agree with Wolfgang's answer, but I'd like to add something of my experience. I had recently to choose between tri and quad based FEM and after googling for it, I decided that I would code both from scratch for a fair comparison (I had a previous experience in writing FEM kernel. I used C++ and the Eigen library for matrices and vectors. I considered only cartesian meshes).
The answer by Wolfgang seemed to hold in my cases (simple advection-diffusion problems) as well, so I can confirm this. However, the "same amount of work" is not exactly true. There are other differences that we need take into account for a true time to error comparison.
- The number of quadrature points might be different. In my case, the tri quadrature had 1 point while the quad one had 4 points. That makes twice less quadrature nodes in the assembly procedure for the tri based FEM. However, this depends on the problem at hand.
- The number of non zero entries in the resulting matrix is different for the two approaches. For a cartesian-like mesh with triangles, a internal degree of freedom typically couples with 6 others, while with quad base, it does so with 8 others. That makes the assembly more expensive and might also affect the linear solver depending on which one is used.
- The linear system for the quad-based FEM was (suprisingly?) easier for the iterative solver than the tri-based (less iterations for a given tolerance). I used an ILUT-preconditioned Krylov-based solver (can't remember which one exactly). I don't know if this is general or specific to the problem, but that is what I saw.
In the end, I still found that quad-based FEM was faster for the time-to-error comparison, but the factor was definitly less than 10, around 3 if I remember correctly (This number of course depends on the specific implementations and problems).
Another point for the quad-based FEM is that it is WAY easier to implement if you want to consider high order or "high" dimensional (>2) FEM.
Hope this is useful!