Hi mwmmboy -- First, of course, I work for Sonnet, but I try my best to give both advantages and disadvantages whenever possible.
I use the term shielded, because "closed" could be misread as though it does not interface to other frameworks. I also use the term unshielded, because "open" could be misread as though it does interface to other frameworks, even in cases where it does not.
Both types use MoM (Method of Moments). This approach divides a circuit (the metal part only) into many small subsections. It then calculates the coupling between subsections (i.e., put current on one subsection, calculate the voltage induced in another subsection). These pair-wise couplings fill a big matrix. The matrix gets inverted and the problem is solved.
Shielded analysis calculates the coupling by a 2-D FFT. A single 2-D FFT calculates all the coupling between all possible subsections of a given type (say X directed coupling to Y directed subsections, etc.) on a given level. This absolutely the fastest way to do it, by far. In fact, it is so fast, there is absolutely no point to store the results of the FFT for later access. It is also the most accurate, the coupling is calculated to full numerical precision. The numercial noise floor (which we have measured in multiple ways) is typically 100 to 180 dB down, most commonly 120 to 140 dB down. (You can not get accurate data once you are within about 20 dB of the noise floor.)
When you use the FFT for signal processing, you must first uniform time sample your signal. Same thing in MoM. Only now, we uniform space sample the surface of the substrate. This is the principle disadvantage of the shieled/FFT approach. Fortunately, you can make the grid size very fine (mesh the substrate 1000 x 1000 cells is no problem, as long as you keep your subsection count under about 20,000). At this meshing level, the cell size is about the size of a pixel on a computer screen.
The unhielded tools use numerical integration. Because you can integrate anywhere you want, your subsections are now not restricted to a grid. They can be any size or shape you want. However, because they use numerical integration, the coupling between subsections is not calculated to full precision. There is numerical integration error. They typically refine the numerical integration with a target of 3 digits of accuracy (including zeros to the right of the decimal place). I have seen results from unshielded tools showing noise floors ranging from 40 to 80 dB down. 60 dB down is pretty typical. In addition, the calculation of the field due to a patch of current ("Green's" function) is somewhat lengthy, but the tools can typically save that calculation to disk for later use. It must be recalculated only when the dielectric stack-up is changed.
Another situation, commented on in the thread a few days ago in this forum, "Doubts about deembedding", might also be of interest to some readers (you have already seen it and added an appropriate comment). Basically, shielded analyses have direct access to a perfect ground reference for thier ports, and thus are capable of exact (to within numerical precision) results. Unshielded tools do not have this capability.
Basically, I always recommend designers use both shielded and unshielded tools, even though my company sells only a shielded tool. However, it is critcally important that the designer be fully aware of both advantages and disadvantages. Anyone who knows only the advantages of only one tool will have problems.
And Dan Swanson's book (recommended above) is absolutely first rate. It should be required reading for anyone doing EM analysis.