Suppose we are trying to decide whether to use an ordinal regression model or a nominal regression model. If we know that the outcome variable is ordered should we still use a nominal regression model to compare with an ordinal model?
There are a variety of ordinal regression models (see Agresti) but they rely on certain assumptions. When those assumptions are violated, the models may become incorrect. The most common assumption is that of proportional odds. Multinomial regression does not make this assumption and can therefore model odds that are not proportional. However, an intermediate model (such as partial proportional odds) may by justified and yet be less complex than the multinomial model.
If the categories have order, one can use either nominal or ordinal. If the categories do not have order, one must use nominal.
When there is order, one should perform model diagnostics to see which of ordinal and nominal is preferred. Although, the ordinal may have better interpretation. In general, for comparison among different models, one should consider both interpretability and goodness of fit.