You just end up looking and feeling stupid when Morrigan knows more about Dalish customs and gods than you do.
She is the daughter of an elven god mind you.
Regarding being an elf, I do find that it's more strongly tied to the main quest than other races: even if you aren't romancing him, the Inquisitor's relationship to Solas runs slightly deeper if he/she's an elf (for instance he'll share more info with you about ancient elves and be more inclusive of you in your discussion about the orb). Furthermore, you have a more personal connection with the Winter Palace mission (even if Briala will only comment on your race once or twice) and a closer link to the Well of Sorrows questline. Though I've not played it yet, Trespasser also sounds more relevant to an elven character, particularly one romancing Solas.
Probably the most interesting part of this game is the discussion about modern elven culture and how it relates with past elven culture (a discussion that resonates with numerous real world cultures and societies seeking distinct identities) - how it has evolved and twisted certain aspects and what your opinion is on that. For an elven Inquisitor, this discussion is a lot deeper than it might be for a human, dwarf or qunari and you get a larger say in it.