There is no difference in meaning but a significant one in usage.

1. Ordonner is strong and what would be used for a demand¹. An alternative would be:

   Le professeur a donné l’ordre de se taire à l’élève.

2. Commander à quelqu’un de faire quelque chose is an outdated, literary way to express the same but unlikely to be used in modern French.

¹ _{Beware that demander just means "to ask", not "to demand".}

There is no difference; “commander (à qqn) de + inf.” and “Ordonner de + inf. (à qqn.)” are synonyms.