Another "Hilbertian" example: Bessel's inequality follows from Bessel's equality. See, e.g., http://www.math.uri.edu/~quinn/web/mth629_Bessels.pdf.
And now (maybe off-topic, but the question is rather vague) an example of an inequality derived via an identity:
The (simple) identity is the so-called "multiplication of means", roughly: the expectation of a product of independent random variables equals the product of their expectations. The (not so simple) inequality is the Grothendieck one: http://www.ams.org/proc/1987-100-01/S0002-9939-1987-0883401-0/S0002-9939-1987-0883401-0.pdf. (Well, it is not obtained from that identity in an obvious and direct way, but the identity is an essential ingredient in the proof.)