SedPHaT makes the same distinction between Run and Fit as in SedfiT. In brief, the Run command executes a model, i.e. it performs a simulation of the sedimentation process or autocorrelation function, using the parameters that were entered in the global, local, and experimental parameters section. It does optimize all linear parameters, but does not optimize the (usually more important) non-linear parameters.
The reason is that for non-linear regression, one needs good starting estimates for the parameters. Without them, the fit can frequently converge in local minima, and the best-fit may not be found. There is no good alternative to just ('manually') exploring different parameter ranges, and position them so that they'll be able to converge to the overall best fit.
In the process of exploring the parameter space, you will need to assess what the current parameter values do to the model. This is done with the Run command. (See also The difference between 'run' and 'fit' in SedfiT.)
For a description of the goodness-of-fit, see here.
A new concept in SedPHaT is that you can execute the Run command for a single experiment only, or for all of them.