There are many similarities between these two poems.  They both refer to an almighty power, either God or a darker power, that controls some of the more merciless aspects of nature.  In "Design," Frost describes a spider that had killed a moth, and attributes the moth's being there, and the spider being able to kill it in that moment, to a "design of darkness" that enabled such an act.  In "Apparantly with no Surprise," Dickinson relates the cruel frost's killing of a flower as happening under the eye of "an approving God."  So, both poems attribute these forces of nature to a greater power than themselves.  Another similarity is that both Frost and Dickinson were moved by these acts of nature--frost's cruel killing of a flower, and a spider's cruel killing of a moth.  They were moved enough to write about it and ponder why such things happen in life.  They are also both rather short; Dickinson's poems is one stanza, Frost's two.  They both use poetic techniques.  One is a simile, where Frost compares the says the event is like "a witches' broth," "a kite," "rigid satin cloth," and "a froth."  Dickinson uses personfication to describe the frost as a "blond assassin."

The main difference in these poems is who each poet attributes these acts of nature.  As seen above, Dickinson attributes them to "an approving God," whereas Frost is more dismal, stating that a "design of darkness" makes such things happen.  So, they are both in awe, but attribute these acts to opposite ends of the spectrum.  That is the main difference to focus on; there are other obvious ones, like one is about frost, the other a spider, but those are too simplistic to dwell on in depth.

I hope that those thoughts help a bit; good luck!