The Duke in Browning's "Duchess" is all about power, control, and ownership. He doesn't care any more about the "last duchess" as he does for the statues and other treasures he possesses. For him, it is all about having things, not for the love of these things, but for the wealth it brings him. He operates more on fear than on love and passion. He indicates this in the interview with the representative for the new duchess...in essence, he says, "Tell her that she will behave as I say she will behave, or she will be my new last duchess hanging here on the wall."
Rosetti's tone is completely different. He seems to truly have loved the subject he has captured in the painting. He mentions the memories of the walks and talks they had which makes him melacholy.
That day we met there, I and she
One with the other all alone;
And we were blithe; yet memory
Saddens those hours, as when the moon
Looks upon daylight.
The speaker speaks of his time with her, how he longs for her, and how the "earth is over her". She is dead, and he misses her. Her alone. He does not speak of other possessions, power, greed, lust for riches or the next duchess as Browning's Duke does.
The speaker in Rosetti's poem misses that woman behind the glass at whom he stares until she seems to breathe and stir. He is mourning her absence. The tone is completely different. His feelings are completely different.