What is the benefit from going from unordered to oredered data? Let us assume that we have naturally ordered data that we want to classify. Then we can use ordinal regression/classification methods. Yet we can treat those as unordered and use multiclass clasiffication. It seems from first glance like the ordinal case modelling is less studied. Could someone lay down the gains that one gets from going from unordered to ordinal?

In general, it's rarely a great decision to apply a transformation or make a simplification that results in the loss of useful information. Such is the decision to treat an ordered response as unordered because we know how to do classification but don't know about ordinal regression (yet).
Intuitively, you gain power as there are more examples to estimate $\operatorname{Pr}(Y \geq y_k)$ than $\operatorname{Pr}(Y = y_k)$. The quantities estimated by the fitted model can be more meaningfully interpreted as well; for example, what's the probability that a customer gives a product at least 4 stars.
PS. Here are some free online resources about ordinal regression: 
٭ The Statistics Tutorials by the UCLA Statistical Consulting team. 
٭ The Resources for Ordinal Regression Models as well as the Regression Modeling Strategies course by Frank Harrell. 
