The Medea of Euripides lacks the keen focus of pure vengeance that Seneca portrays in his version of her. Of course, Euripides must eventually get around to the theme of vengeance and the fury of a woman scorned—this is, after all, one of the main themes of Medea's mythology. But whereas the Euripides version has her enter the play bemoaning the fate of women, who must rely on the moral rectitude of their husbands in order to achieve happiness, the Medea of Seneca wastes no time at all, opening the play with a clear, haunting statement of her white-hot rage and desire for vengeance. The Euripides Medea is also a bit more calculating than Seneca's version—she makes clear plans to escape to Athens when she secures an oath from Aegeus to give her sanctuary in the future. Seneca's Medea, on the other hand, does not seem to have any future plans beyond the taking of vengeance, and she only escapes due to a deus ex machina—the appearance of a serpent-borne chariot, which bears her away to apparent safety.
The main difference between the two Medeas is that Seneca's version is much more of an active character. There's no weeping or wailing at Jason's abandonment; just a determined, full-throated cry for vengeance. At the same time, this makes Seneca's Medea much more passionate, less coldly calculating than she appears in Euripides. In Euripides, Medea's misfortunes are related to us by her nurse, and we only hear her plaintive cries of injustice from off stage. This establishes a distance between Medea and the audience, allowing her to come across as less human and more like a monster. Perhaps this is a way for Euripides to make the horrific acts that are about to follow more readily understandable. After all, it's marginally less unpalatable to be presented with Medea's unspeakably barbarous acts if they're committed by some kind of scheming monster, rather than as being the knee-jerk response of a wronged wife and mother.
Seneca's Medea is more godlike, whereas Euripides's is almost a plaything of the gods. In the former, she's a force of nature, taking control of events. She even occasionally has the audacity to curse the gods for their inaction on her behalf. But they ultimately save her in both versions of the story, though with one subtle difference. In Euripides, Medea is already in Helios's chariot when it appears; in Seneca, the chariot only appears after Medea has summoned it. Again, she's the one who's in control of her fate.
This is a great question. There are many differences. So, I will only mention a few.
First, it is important to ground these two plays in history. Euripides in writing for the Athenians in the 400s BC and Seneca is writing in Latin for the Romans during the first century AD under Nero. Some 500 years separates the two works.
Second, if we start with Euripides' work, Medea comes off as a tragic figure. She is strong, but she is also a victim. We are able to sympathize with her plight and there is a sense of pathos. We might even be tempted to say that she is an unfortunate victim of the gods or fate. When we look at Seneca's play, we feel little of these things. Medea right from the beginning is a frightful figure. She immediately is filled with rage and plots her revenge. The start of the play says it all. In Euripides account, Medea speaks of her suffering; in Seneca's she prays for revenge.
Second, the relationship between Medea and the gods also differs. In Seneca's account, the gods are challenged and even cursed; in Euripides' account, Medea capitulates.
We can look at Seneca's version as a retelling, since Euripides' Medea was well-known in the ancient world.