The difference is not big.
I would argue that the semantic difference between philosophia naturalis and philosophia naturae in Latin is the same as between "natural philosophy" and "philosophy of nature" in English.
I find the exact difference between the two to be a matter of taste, at least to an extent.
The genitive in philosophia naturae is an objective genitive, just like in amor naturae the love is felt towards nature rather than owned by it in any sense.
Therefore the genitive describes the object of philosophical study.
The adjective naturalis describes the philosophy rather than its object.
Philosophy can be natural in many ways: it can exclude fictional things and focus on what is real, it can be the kind of thought that arises naturally in a human being, it can study nature, or it can be related to nature in some other way.
The two overlap, but it seems to me that philosophia naturae is a little more narrow than philosophia naturalis.
In general, I would regard this simply as a stylistic choice.
If an author uses both terms and defines a difference between them, then the difference is what it is described to be.
Language alone does not make a clear enough distinction between the two.
There might be a canonical distinction between natural philosophy and philosophy of nature, but that would be within the realm of philosophical canon rather than linguistic canon.
Latin has a tendency to use derived adjectives over the genitive, at least in comparison to English.
If the author is not trying to make a distinction, this will introduce a language-dependent bias on what is likely to be chosen.
If, however, the author wants to make a distinction between the two, then I think the differences are the same in English and Latin.
Some derived words have obtained meanings and nuances that are no longer directly linked to the original word.
For example, moralis is perhaps not adequately described as just "related to mos or mores".
When that happens, the change of connotation brings about a difference between philosophia moris and philosophia moralis.
The difference between these two is not well explained by any general theory comparing genitives and derived adjectives.
What you need to know is the meanings of the noun and the adjective, and one cannot be safely deduced from the other.