A physician wrote hydrocortisone in a mouth gel for a child. This should not be hydrocortisone acetate.

The TMF used for the oral gel hydrocortisone acetate. In oral preparations, the flavour is important. Hydrocortisone acetate is less soluble in water than hydrocortisone and thus a possible taste will be lower. It is counted on a reasonably long stay of the gel in the mouth.  

The difference in choice for a mouth gel or for a mouthwash I cannot explain. You do not mention the remaining constituents of the mouthwash; Also not these from the mouth gel. Answering questions with very little information is difficult not to say impossible.