The Duke does not really understand or appreciate art. His interest in art is that of a collector and an investor. He makes a point of telling his visitor that the portrait of his wife was painted by a fictitious artist called Fra Pandolf knowing the name of the artist will impress the other man. The Duke does something similar when he mentions Claus of Insbruck, also a fictitious artist but one whose name should impress anyone acquainted with contemporary artists. Like many modern art collectors, the Duke knows that art is a good investment if the artists' names are well-known, e.g., Van Gogh, Picasso. Also, like many art collectors, the Duke enjoys the ability to show his collection to other people. He even enjoys showing his dead wife to other people, judging from his monologue, even though he admits that he had her murdered. Browning takes pains to show that the Duke is vulgar and ignorant in spite of his aristocratic status. The Duke would have to have a coarse nature to behave as he did.
The Duke prizes the portrait of his wife exactly because it is a work of art. He did not prize his wife - he had her murdered. But now she is a work of art, like the the bronze statue of Neptune taming the sea horse (how ironic!!!) she is valued and loved.(He could not tame her!!!)
The Duke only loves things that he can control, like art or the artists who rely on his patronage. In life he was unable to command her complete attention. She was too ready to rank the gift of his noble name with those things given by others. She smiled at everyone, now she only smiles at his behest when he draws the curtain aside to control who she smiles at.
The language reflects this as well as the narrative. When he speaks about art or artists the tone is controlled but when he begins to talk about the Duchess when she was alive, the lines run on (enjambment) as if out of control and the pauses (caeasura) indicate a further loss of control.
In short the Duke loves art because it is under his control.