The newer right seems to be more geared toward infant baptism, I am assuming that in the case of baptism of an older child or an adult convert, some modifications are already incorporated into the rite.
The other thing I notice is that the newer rite contains much more cathecesis. One of the shifts that came along in the wake of V2 was, at least as far as this 39 year old can tell, was an increase in cathecesis. Holy Mother Church wants her children to better understand her teachings, which are those of her Lord, Jesus Christ.
It appears to me that the attitude before the council was that most lay Catholics should simply attend Mass on Sundays and Holy Days and observe the other precepts, but that understanding was not that important. I think that many viewed the Latin language as a sort of temple veil, so that the unclean masses would not get to close to the sacred truths. The belief was that this distance would enhance the mysterium fidei, and thus reverence for what is holy.
In the post-conciliar Church, there is an increased focus on cathecesis of the laity, a desire that they too can understand what the Church teaches and why she teaches it. For me, personnally, the more I learn about the mystery that is the Catholic faith, the more I realize I do not know, and can never fully understand, therefore, increasing my awe and wonder at this glorious and supernatural faith that Christ has given us.