grammaticality - A friend of John's / John's friend

The question:

Suppose John is my friend, and I am introducing myself to his brother, should I say

"I am a friend of John's"

or

"I am John's friend"

??

I would use the former, but some think the latter is also correct. If both are acceptable, what are the differences? If we can already say John's friend, why do we need constructions like a friend of John's in English?

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^Note:

<sup>There are lots of question on this site asking about a friend of John versus *a friend of John's
* and so forth. This however, is not one of those questions!</sup>