There are many SSPXer's here who will tell you that the Novus Ordo is bastardized and illegal, since Pius V said the Tridentine Mass was to be celebrated for "time immemorial." ` They also don't like the Novus Ordo because it's in the vernacular language, the priest faces the people, and a host of other changes from the Tridentine Mass.
No Pope can bind another Pope on matters of liturgy, so the Novus Ordo is the manner of celebrating the Mass used today.
The SSPX left in a huff in 1987, ostensibly over the Tridentine Mass, but really over the fact that its members didn't accept Vatican II.
The Tridentine Mass is permitted today, but only with special permission from the bishop..
The so-called Tridentine Mass is that rite of Mass promulgated by Pope St. Pius V in his 1570 declaration Quo Primum. In promulgating this rite, St. Pius V was following the directives of the Council of Trent (the Latin word for Trent is "Tridentum" -- that's where the name comes from), which hoped to better maintain Church unity through a uniform practice of worship. Nevertheless, St. Pius V allowed the co-existence of a few other rites (e.g., the Dominican, Ambrosian, Carthusian, Oriental, etc.) because they had been in continuous use for over 200 years.
The rite of Mass that St. Pius V applied to the whole Church was actually the rite of Mass as then celebrated in the Diocese of Rome. In fact, in all its essentials, it was the way that Mass was celebrated in Rome by the 4th Century. Therefore, it is better to call it the "Classical Mass" rather than the "Tridentine Mass," since its pre-dates the Council of Trent by many centuries.
In October of 1962, the Fathers of Vatican Council II re-examined the rite of Mass in response to long-standing criticisms of some of the complexities that had come to mark Catholic rituals over the centuries. The Fathers suggested that some minor changes be made to the rite of Mass, and that some vernacular language could be used in place of Latin if pastoral reasons so indicated.
The rite of Mass produced by Pope Paul VI's Consilium in 1969 bore little or no resemblance to the Mass envisioned by the Fathers of Vatican Council II. In essence, while the Vatican II Fathers had endorsed the idea of organic (i.e., slow, well-considered) changes naturally following the progress of previous developments in the rite, Pope Paul's Consilium produced an entirely new rite out of whole cloth. Worse yet, this new rite was created by a committee of bureaucrats.
Recent statements by confidantes of Paul VI suggest that he intended his new rite as an ecumencial sop to the most radical of Protestants. But, of course, the Prots weren't interested in window-dressing -- they wanted doctrinal change, and, even under Paul VI, they weren't gonna get it. The gates of hell won't prevail. Rather than satisfying the most radical of Protestants, Pope Paul seems only to have satisfied the most bourgeois of bourgeois Catholics, since the "New Rite," in all its bland, non-threatening comfort poses no intellectual or spiritual challenge to even the most indolent of the couch potato set.
Even non-Catholic artists and intellectuals were incensed with the New Rite, seeing in it a triumph of mediocrity over beauty. To compare the Classical Rite with the New Rite is to compare the Pantheon with a suburban strip mall.
The validity of the New Rite cannot be questioned -- after Paul VI heeded Cardinal Ottaviani's intervention that in its original Consilum-produced form, it was doctrinally unsound. On the other hand, the manner in which the New Rite was imposed on the Church and the outrageous mistranslations of presently used vernacular texts from the Latin are two of the great scandals of Christian history. And while Archbishop Lefebvre certainly should not have consecrated new bishops and provoked a schism, the brutal, dishonest, and back-stabbing treatment he received from both Vatican functionaries and the French bishops certainly mitigates his moral culpability.
In sum, the New Rite can be likened to a tragic pregnancy. In its conception, the New Rite is like a Down's Syndrome baby. In its imposition on the Catholic world, it is like a disastrously malpractice-ridden breech delivery. With the exception of those rare places where it is more-or-less celebrated according to the rules, the New Mass is like a Down's Syndrome child with several neurological palsies secondary to a problem delivery.