That is a very interesting question. I would say that one similarity is that both stories focus on young protagonists. Romeo and Juliet are both teenagers. We are not told Romeo's exact age, but Capulet does state in Act I, Scene 2 that Juliet isn't quite 14 yet:
My child is yet a stranger in the world.
She hath not seen the change of fourteen years.
When The Chrysalids begins, readers are told that David is ten years old:
That day I had gone off by myself, as I often did. I was, I suppose, nearly ten years old. My next sister, Sarah, was five years older, and the gap meant that I played a great deal alone.
The book takes place over the course of years, and David is around the age of sixteen by the time the novel ends. Readers know this because chapter eight tells readers that six years have passed from the time that David was introduced to readers:
Somehow, through caution, luck, and quick recoveries we managed to escape direct suspicion and live our two diverging lives for the next six years without the sense of peril becoming sharp.
I see similarities between a few of the parents in both stories as well. Both David and Juliet have very domineering fathers. It's their way or nothing. Consequently, Juliet's mother and David's mother are both weak and submissive characters. Emily Strorm does not do a thing to stop David from receiving a beating from his father, nor does Lady Capulet argue with her husband's decision to force Juliet to marry Paris.
A third similarity is that both stories have violence. Romeo and Juliet has sword fighting and duels, and The Chrysalids has battles with weapons like bows and arrows.
A major difference between the two stories is that the main protagonists in Romeo and Juliet die by committing suicide. While there is suicide in The Chrysalids, and main characters do die, David and the other main protagonists escape the forces threatening their lives and survive.
A second major difference is a general genre difference. Romeo and Juliet is a tragedy with a heavy dose of romance. The Chrysalids, on the other hand, is science fiction. I suppose an author could write a science fiction tragedy, but the story is so full of action, suspense, and post-apocalyptic elements that there's very little time spent on actual romance. There are sad moments that occur in The Chrysalids, but that doesn't make it a tragedy. While the ending is a bit of a cliffhanger, it is definitely a happy ending to the story. The same cannot be said about Romeo and Juliet.
Similarities: Both works center upon forbidden love. In Romeo and Juliet, the title characters cannot freely and openly express their love for one another because of their feuding families. In The Chrysalids, David and Sophie cannot openly love one another because David's father would not approve of his love for a girl who is different and unlike their strict "Old Testament" culture.
Likewise, the play and novel use teens as their main characters--teens who often appear to be wiser than the works' adults. Yes, Romeo and Juliet are impulsive, but they are also wise enough to realize that the long-standing feud is pointless. David, too, realizes that he and the other "telepathic" teens seem to have a better grasp of how life should really be lived.
Differences: Romeo and Juliet's lead characters commit suicide, and their tragic ending teaches the adults in the play a moral. While there is suicide in The Chrysalids, the main characters do not take their own lives, and in the end they remain with a message of hope for the readers, while the perpetrators of most of the novel's conflict are eliminated.
Another obvious difference is the setting of each work. While Shakespeare's play follows typical Renaissance customs and plotlines, The Chrysalids is a very modern post-Apocalyptic tale.
David and Sophie were actually not in love with each other. They were just really good friends. David has always loved Rosalind.