Both of these stories highlight one of Mark Twain's greatest talents, which is the use of dialect. Dialect is when an author writes down the way people speak in exactness, complete with accents, slang, idioms and local color. In both of these stories, dialect is a main feature, one that allows the reader to feel like he's been transported back in time to these southern places where people talk a bit differently. So, dialect is one commonality.
Another way they are similar is that they use satire to make fun of certain types of people. Twain loved making fun of people in his writing, and did so in these stories through telling tales of the people that were exaggerated, in order to highlight how absurd they were. For example, in "The Celebrated Jumping Frog of Calaveras County," the protagonist, Jim Smiley, bets on anything and everything. The story is exaggerated to emphasize the indolent and gambling lifestyle of certain types in the south in that time period. In Huck Finn, Twain includes tons of stories that exaggerate people in order to make fun of them; consider the intense feud, the circus, the episode with Boggs, and many of the other absurd tales. So, satire and poking fun at others is a common element of both stories.
The settings are also similar; both have major portions taking place in the south. Main characters also show similar traits; the Duke and the King are similar to Jim Smiley in their insatiable need to make money and take advantage of others to do so.
I hope that those thoughts helped; good luck!