There are actually quite a few similarities between the two. Both couples were separated by family politics, but were able to get around them somehow (the crack in the wall for Pyramus and Thisbe; the masquerade and subsequent balcony scene for Romeo and Juliet) . They both agreed to marry in spite of their parents' disapproval, and both female leads attempted to contrive some way to be together with their lover in spite of their circumstances (Thisbe was scared away by a lioness with jaws dripping blood, and she left her shawl behind which the lioness chewed up; Juliet put herself into a death-like sleep in the hopes that she would escape her own impeding arranged marriage). However, everything went awry when the male leads thought that their beloved had been lost to them forever (Pyramus saw the shawl, and stabbed himself with this sword; Romeo saw Juliet in her deathlike sleep, and drank poison). The suicide of the male leads was soon after imitated by their female counterparts (Thisbe stabbed herself with Pyramus' sword; Juliet stabbed herself with Romeo's dagger).
In plot and, to a certain extent, theme, there really isn't much difference between the story of Pyramus and Thisbe and Romeo and Juliet - it is even possible to assume that Shakespeare derived inspiration for Romeo and Juliet from the story of Pyramus and Thisbe (Pyramus and Thisbe is a story of Roman origins). Shakespeare only added more characters to the story, emphasized the family rivalries, and set his story in Verona.
