In practice, this difference is obvious, but how to put it in words?
I would describe hyper parameters as "indirectly learned/trained". Think of the inference chain.
Data --> Parameter(s) --> Hyper parameter(s)
The hyper parameters are not directly connected to the data observed.
Another way is to consider a scenario where you know the parameter(s). Is there any point using the data? If the answer is no, then it's a hyper parameter.