The Greek myth of "Cupid and Psyche" obviously came before the Grimms' "Frog King." As a myth, the story of "Cupid and Psyche" is meant to entertain and reaffirm the power of the gods in a way that is perceived as a true event. "The Frog King," however, is written not as a truth, but to entertain and instill a motifs.
The first similarity is in the first paragraph of both tales. Both introductions revolve around the youngest daughter who is beyond beautiful and adored by all.
~ "In olden times, when wishing still did some good, there lived a king whose daughters were all beautiful, but the youngest was so beautiful that the sun itself, who, indeed, has seen so much, marveled every time it shone upon her face." - The Brothers Grimm (ed 3)
~ "A certain king and queen had three daughters. The charms of the two elder were more than common, but the beauty of the youngest was so wonderful that the poverty of language is unable to express its due praise. The fame of her beauty was so great that strangers from neighboring countries came in crowds to enjoy the sight, and looked on her with amazement, paying her that homage which is due only to Venus herself." - Lucius Apuleius
The biggest similarity between the stories is that both tales hold true to the Beauty and the Beast motif.
Not only are Psyche and the Grimms' princess companions to beasts (the unknown figure of Cupid and a frog), but both tales exemplify the beasts within the princesses themselves. This is shown in "Cupid and Psyche" when Psyche looks upon her husband for the first time, wielding a knife, prepared to kill him if he turns out to be the gruesome monster her sisters foreboded. Consequently, in "The Frog King," the princess shows her beastly side as she angrily throws the frog against the wall, in hopes of killing him.
That's about where the similarities end.
"Cupid and Psyche," being a Greek myth simply has more: more elaborate detail, more characters (as many of the gods fill their own part), and a much more complex plot structure. The plot of the stories themselves are completely different (and I'm not going into the entirety of that detail).
A specific difference (worth making note of) is the relationships between Psyche and Cupid vs that of the princess and the frog.
Although Psyche is initially fearful, crying at the top of the mountain waiting for her future husband, her mindset is immediately reversed upon receiving the benefits of her new lifestyle. These feelings then transpose to her unseen husband, whom she truly loves.
As you can see these stories are quite different, but they hold a common base.
Bottom line: Whether it's "Cupid and Psyche," "The Frog King," or even Disney's movie "The Princess and the Frog," stories have been and always will be appropriated in one way or another.