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Compared to what? The adavantage is they are much faster than calculating with an abacus. The disadvantage is that they cost more than an abacus. They are tools. It's like asking what are the advantages and disadvatages of shovels and hammers. Are you asking about when to use one or the other?
HFSS is full-wave 3D which means it solves all of Maxwell's Equations. The limit on accuracy is basically dependent on the mesh size. HFSS is very good at extracting s-parameters and fields.
Q3D is quasi-static 3D which means it dosesn't solve the time dependence piece of Maxwell's equations. Essentially you want the EM wave to have a wavelength of about 8-10x larger than the structure you are simulating. This essentially will ensure the quasi-static approximation holds true. Q3D is very good at extracting R, L, C, and G parasitics.
Both HFSS and Q3D use automated adaptive meshing to converge to the desired accuracy of the solution. Speed varies depending on the structure and the usage of compute resources. For instance HFSS can distribute frequency points over a cluster which will dramatically speed it up (say 10-20x) where as Q3D can distribute the CG, ACRL, and DCRL solvers over a cluster and then use MPI for CG frequency points. In general different q3d applications can see speedups ranging from 10x to 100x on a large cluster.
In the end the best tool depends on your application. You should answer the following: Do you want s-parameters or do you want RCLG parameters?