I've always used "inherent" and "intrinsic" interchangeably. Dictionary.com doesn't offer much help in distinguishing them.
I do not know about English usage, but coming from Latin there are some differences:
When trying to distinguish between nearly synonymic words, dictionaries - because they try to encompass all usages of the term - yield poor discriminability.
It is better to look at how the terms appear in actual use to flush out the subtle differences in the minds of the users. As a rough first pass using the "compare" feature of the Corpus of Contemporary American English can give a picture of collocates to your two words of interest. If the words lived in the minds of speakers as wholly interchangeable you'd expect to see nearly identical collocates, in this case they don't.
Unfortunately, that site doesn't produce nice "click me" links, so go there, pick Display > Compare, and put your two words in the two "search string" boxes just below. The rest of the defaults are fine, and press search. And then click around on various bits.
The words are used in pretty disjoint contexts therefore they aren't precisely interchangeable, however defining the relevant contexts is a toughy. For everyday speech, most listeners will not balk at using them as if they were interchangeable.
New Oxford American English:
A quality that is inherent is a permanent part of a person's nature or essence (: an inherent tendency to fight back).
Intrinsic and essential are broader terms that can apply to things as well as people. [...] an intrinsic quality is one that belongs naturally to a person or thing (: her intrinsic fairness; an intrinsic weakness in the design).
So theoretically inherent has to do with a person and intrinsic with either a person or thing but...that's REALLY nitpicky. In fact, the very definition of inherent, according to the same dictionary is
inherent |ɪnˈhɪrənt| |ɪnˈhɛrənt|
existing in something as a permanent, essential, or characteristic attribute : any form of mountaineering has its inherent dangers | the symbolism inherent in all folk tales.
Merriam-Webster considers them synonyms. So it's really up to whatever you choose to believe. I doubt most people would perceive any difference, though.
The uses of these two words definitely overlap, but I do think that, at times, there is a subtle difference.
In my experience, intrinsic is more frequently used when the property is unique or unexpected, while inherent is more frequently a property that would be expected or common. Ergo, intrinsic implies a certain uniqueness that inherent does not.
I also believe intrinsic is also more frequently used to refer to properties of nature-related subjects or naturally-occurring properties, for instance, a precious metal, a mountain view, etc. In this sense, an intrinsic property would be considered a "universal truth" for that subject, and would be true for any instance of the subject. Still, for intrinsic, the property is somewhat special or unique, per my first statement.