What are the similarities and differences between "How do I Love Thee" by Elizabeth Barrett Browning and "The Last Duchess" by Robert Browning?

There are more differences than similarities between the two poems. The chief similarity is that they are both about love (or what a character in a poem thinks love is). One could argue that the adoring love expressed in "How Do I Love Thee" is what the duke wanted from his late duchess.
Otherwise, the two poems are very different. First, "How Do I Love Thee" is a sonnet that directly addresses the beloved, whereas "My Last Duchess" is a dramatic monologue in which the duke is telling a courtier assigned to contract a marriage for him the story of his deceased wife.
Second, the tone of the two poems is very different. The tone of "How Do I Love Thee" is sincere and heartfelt, an expression of pure love and devotion. There is no underlying agenda beyond the expression of romantic love. "My Last Duchess," on the other hand, is ironic, in that the duke betrays the opposite of what he intends. He wants to cast blame on the young duchess for not being devoted enough to him, but he ends up looking like a sociopathic monster.
Third, the subject matter of "My Last Duchess" is much darker. This is a narrative poem, meaning it tells a story, whereas "How Do I Love Thee" is simply a lyrical (or beautiful) expression of love. In "My Last Duchess," it becomes clear that the jealousy the duke experienced led him to have his sweet young wife killed.
Both poems are about love, but one is about a pure and sincere love while the other conveys a very twisted view of what love means to a disturbed duke.