Seneca's Medea differs from Euripides' Medea in that Euripides' feels the horror of her act but is overwhelmed with raging emotions against humiliation and against letting someone else have and raise her children while Seneca's is emboldened and vengful and full of hardening hatred. Euripides' Medea is a more complex, one might say more human, character.
Seneca's version of this classic presents Medea as the ultimate femme fatale and as somebody who is defiant about what she has done and has no regrets or grief. The figure of Medea is one that bristles with anger and seems to deliberately antagonise both the audience and those around her. The Medea of Euripedes, on the other hand, is a very different figure as is presented much more sympathetically as a character who is caught up in the fate and the will of the gods. Her actions are presented as being the natural outflow of such a situation.
Two passages from the plays will help make the contrasts clear. In the play by Euripides, here is the final speech before Medea goes off to kill her children; obviously she is tortured by the mere thought of what she is about to do:
My friends, I am resolved upon the deed; at once will I slay my children and then leave this land, without delaying long enough to hand them over to some more savage hand to butcher. Needs must they die in any case; and since they must, I will slay them-I, the mother that bare them. O heart of mine, steel thyself! Why do I hesitate to do the awful deed that must be done? Come, take the sword, thou wretched hand of mine! Take it, and advance to the post whence starts thy life of sorrow! Away with cowardice! Give not one thought to thy babes, how dear they are or howthou art their mother. This one brief day forget thy children dear, and after that lament; for though thou wilt slay them yet they were thy darlings still, and I am a lady of sorrows.
These are the very last words she speaks, and her killing of her children takes place off stage and is reported, very briefly, to Jason at the end of the play. In contrast, here is the comparable moment in Seneca's play; notice how vicious and bloodthirsty this Medea seems as she slays her children not only on stage but in front of her husband, and notice how she glories in her deeds:
See, there she is herself, leaning over the sheer battlement! Someone bring fire that she may fall consumed by her own flames.
 Nay, Jason, heap up for thy sons their last funeral pyre; build them a tomb. Thy wife and father have already the services due the dead, buried by me; this son has met his doom, and this shall suffer like fate before thy eyes.
 By all the gods, by our flight together, by our marriage couch, to which I have not been faithless, spare the boy. If there is any guilt, ‘tis mine. I give myself up to death; destroy my guilty head.
 Here97 where thou dost forbid it, where it will grieve thee, will I plunge the sword. Go now, haughty man, take thee maids for wives, abandon mothers.
 One is enough for punishment.
 If this hand could be satisfied with the death of one, it would have sought no death at all. Though I slay two, still is the count too small to appease my grief. If in my womb there still lurk any pledge of thee, I’ll search my very vitals with the sword and hale it forth.
 Now end what thou hast begun – I make no more entreaty – and at least spare98my sufferings this suspense.
 Enjoy a slow revenge, hasten not, my grief; mine is the day; we are but using the allotted99 time.
 O heartless one, slay me.
 Thou biddst me pity – [She slays the second son.] ‘Tis well, ‘tis done. I had no more atonement to offer thee, O grief. Lift thy tear-swollen eyes hither, ungrateful Jason. Dost recognize thy wife? ‘Tis thus100 I am wont to flee. A way through the air has opened for me; two serpents offer their scaly necks bending to the yoke. Now, father, take back thy sons. [She throws the bodies down to him.] I through the air on my winged car shall ride.
[She mounts the car and is borne away.]
As you state, there are two versions of the play. Euripides makes Medea a much more likable character. The audience is able to sympathize with her. She is portrayed as a victim of fate or the gods. We might even be tempted to say that we might dare such an act, if we were put in the same situation.
In Seneca's version, Medea is a frightful figure, filled with anger and vitriol. She plots a gruesome revenge and even challenges and curses the gods. In Euripides, she just capitulates to her fate.
In these ways, the audience is much more sympathetic to Euripides's Medea.