Beauty is in the eye of the beholder.

Something that I consider beautiful and is art:

A painting called "The Swing" by Jean-Honoré Fragonard. The painting is beautiful in the sense that it has aesthetic appeal and provides a perceptual experience of pleasure. But the painting can also be classified as art because it is intentional and expressive. There is a deep meaning behind the picture as well: the young man hidden in the bushes is watching the woman on a swing, being pushed by her husband. Her husband is hidden in the shadow, as he is unaware of the affair. Obviously, the artist put a great amount of thought into this painting.

Something that I consider beautiful that is not art:

The Prague Castle. 

Beauty is a very subjective matter (as the old saying goes: "beauty is in the eye of the beholder"). I personally believe that the Prague Castle contains a certain aesthetic appeal that provides a perceptual experience of pleasure. However, the Prague Castle was not originally built for aesthetics. Most castles were built as fortresses from invasions, not to be looked at as works of art. They were functional, as opposed to expressive. So although this castle is beautiful, it is mostly classified as non-art.

Something that I don't consider beautiful but is art:

The painting "The Potato Eaters" by Vincent Van Gogh.
To me, this painting doesn't exactly hold any sort of aesthetic appeal. The colors are a bit dark and depressing, the shapes of the peoples' faces appear odd and disjointed, and the overall picture just isn't pleasing to me. However, the painting was made intentionally by Vincent Van Gogh, and it is both expressive and creative. Therefore, it is art.