
It is almost impossible to predict how difficult it is to solve a particular model. The best and most reliable way to find out which solver to use is to try out both. However, there are a few rules of thumb:
CONOPT is well suited for models with very nonlinear constraints. If you experience that MINOS has problems achieving feasiblity during the optimization, you should try CONOPT. On the other hand, if your model has few nonlinearities outside the objective function, MINOS is probably the best solver.
CONOPT is has a fast method for finding a first feasible solution that is particularly well suited for models with few degrees of freedom (this means: the number of variables is approximately the same as the number of constraints  in other words, models that are almost square). In these cases CONOPT is likely to outperform MINOS while for models with n >> m (many more variables than equations) MINOS is probably more suited.
CONOPT has a preprocessing step in which recursive equations and variables are solved and removed from the model. If you have a model where many equations can be solved one by one, CONOPT will take advantage of this property. Similarly, intermediate variables only used to define objective function terms are eliminated from the model and the constraints are moved into the objective function.
CONOPT has many builtin tests (e.g. tests for detecting poor scaling). Many models that can be improved by the modeler are rejected with a constructive message. CONOPT is therefore a useful diagnostic tool during model development even if another solver is used for the production runs.
It is almost impossible to predict how difficult it is to solve a particular model. However, if you have two solvers, you can try both. The overall reliability is increased and the expected solution time will be reduced.
On a test set of 196 large and difficult models, many poorly scaled or without initial values, both MINOS and CONOPT failed on 14 models. However only 4 failed on both MINOS and CONOPT. So the reliability of the combined set of solvers is much better than any individual solver.
Many examples of poorly formulated models were observed on which MINOS failed. CONOPT rejected many of the models, but with diagnostic messages pinpointing the cause of the problem. After incorporating the changes suggested by CONOPT, both MINOS and CONOPT could solve the model. Switching between the two solvers during the initial model building and debugging phase can often provide useful information for improving the model formulation.
In order to encourage modelers to have two NLP solvers, GAMS offers a 50% discount on the second solver when both MINOS and CONOPT are purchased together.