St. Barts or Anguilla
 

Which would everyone recommend for a week in June for a young couple (early 30s) looking for luxury, good food and relaxation, but also a little bit of culture? Is St. Barts significantly more expensive?

Go to Anguilla if you desire being on a flat arid island with long strands of sugar white sand beaches, some very good restaurants but little, if any, night life, no shopping (other than for t-shirts and trinkets and an overall "Caribbean feel". Pick St. Barts if you want a hilly island, smaller beaches with white to golden white sand, the ability to sunbathe topless, somewhat more natural vegetation, scores of gourmet "French" restaurants, lots of watersports, chic sophisticated shopping, an active nightlife and a decidedly "European" feel. Both islands have plenty of luxury accomodations and you'll probably encounter more people on St, Barts.

Are the beaches appreciably better on Anguilla compared to St. Barts?

Anguilla's beaches are some of the best in the world. St. Bart's beaches don't even come close.

It depends what you mean by better. The beaches on Anguilla are, for the most part, longer and the sand is whiter. The beaches on St. Barth's are shorter (move "cove like") and you'll find more palm trees and other vegetation than you will on Anguilla. Can't say one is better, just different. Most people wouldn't be disappointed in the beaches on either island.

We've been to both islands a couple of times, and Anguilla's beaches are by far the best -- long, broad, glistening white sand. The only beach on St. Bart's which comes close is Flamands. There is a wonderful high-end resort right on Flamands beach that we highly recommend, St. Bart's Ile de France. Both islands have great food. As "A Traveller" wrote, they have a very different feel, but you can't go wrong with either.