If you read the dictionary definitions of the words intrinsic and inherent, you will probably come to the conclusion that they mean completely the same. I beg to differ. While the two words are very similar in meaning (and often interchangeable), there is a subtle difference between them.
The first obvious difference is that they are used with different prepositions: “intrinsic” is used with “to” and “inherent” with “in”, for example,
Helping our friends is intrinsic to our human values.
But what if they are used without a preposition? Is there a difference between an intrinsic property and an inherent property of something?
In my humble opinion, the difference is that if a property is inherent, it is both inseparable and characteristic (that is, important or salient), whereas an intrinsic property, while also inseparable, is usually somewhat secondary, subtle, hidden, or surprising. That’s why we usually speak of an “inherent weakness”, as in
The weakness is inherent because it is unavoidable and important; a doctor performing such a test must be aware of it. On the other hand, we speak more commonly of “intrinsic beauty” rather than “inherent beauty” (although both expressions make perfect sense). A mathematician could say,
By choosing the word “intrinsic”, the speaker shows that he thinks of beauty as of something that is part of mathematics while not being one of its salient features. To say that “a formula shows the inherent beauty of mathematics” would be a stronger statement. It implies that beauty is present everywhere in mathematics; that you will inevitably find beauty if you study mathematics.