Reading in reverse order
It's been entertaining to track the differences between the two. Much is due to the 2,000 year separation of the works, but I tend to read anachronistically. Books that appear in my reading space about the same time become part of the same conversation. I should probably work some Samuel R. Delany in here, something else I'm reading now.
Racine is interested in abstractions like virtue and honor, Euripides deals more directly in natural experience, like the nuances in the connection and rivalry between father and son across generations.
Euripides is bleaker than Racine, no surprise. At the end, Thésée is just left with grief. In Racine, there is the consolation of reconciliation with Aricie and the Pallantides.
Racine added the character of Aricie. This changed the structure in several ways. For one, in Euripides Hippolytus never succumbs to Cypris (Aphrodite) and gets punished for his resistance. In Racine, Hippolytus has lost his resistance and gets punished for his insistence on behaving honorably and not spilling the beans on Phèdre when he first had a chance. See, here's that motive force of an abstraction in Racine--he is undone by acting honorably. In Euripides it's more visceral--the goddess is jealous.
Well, really Hippolytus is punished because he's a douche. His sense of self-righteousness, and his willingness to tell people about it ("there is no man alive/whose wisdom and restraint surpass my own" 1114-1115), earn him no slack when things go wrong. His servant wanrs him early on--"those who act superior are hated" (line 110). His dad is quick to assume he's lying.
Gods exist and play an active role in Euripides. They are almost completely metaphorical in Racine.
The addition of Aricie creates a diamond structure in Racine: Thésée-Phèdre-Hippolyte-Aricie. Thésée loves Phèdre (or at least is connected to her), Phèdre loves Hippolyte, and Hippolyte loves Aricie. Aricie is connected to Thésée by the negative antipathy between the houses. In Euripides there are two triangles: Theseus-Pheadra-Hippolytus, and Cypris-Artemis-Poseidon.
Words don't seem to stand out as much against the backdrop of the action and the play. Some of this may be the fact that I'm reading this in translation, although Diane's verse is excellent. Even so, it seems like the characters don't wield the words the same way.
Lots of dialogue, some rapid and continued back and forth, like lines 331-374, where Phaedra reveals her desire for Hippolytus as she trades single lines of dialogue back and forth with the nurse. This really brings out the monologic quality of Phèdre. The fact that it has monumental stature in French culture must say something about the French. The role of Phèdre was one upon which Bernhardt built her fame. These big chunks of verse would provide a showcase for an actor.
The written word plays a critical role in Hippolytus and has no presence in Phèdre that I recall. Where the critical turn in the plot in Racine comes from false oral testimony, in Euripides it takes the form of writing on a wax tablet Phaedra attaches to her wrist as she is committing suicide. (Diane's translation comes with thorough and incredibly helpful notes, such as the one explaining this wax tablet.)