We'll use Humanae Vitae to explain. In 1931, Pius XI published Casti connubii, which reiterated the clear and consistent Tradition of the Church that because the PRIMARY end of marriage was the having and raising of children, all methods of contraception were never permissible, as they are intrinsically immoral. In 1968, Paul VI "shocked the world" by issuing Humanae vitae, which seemingly upheld the Church's teaching on marriage and contraception. In fact, Paul VI made an almost imperceptible change: marriage now had TWO equal and primary ends: children AND "mutual help".
On the surface, this isn't a big deal, right? However, this slight change has led to a HUGE change in what Catholics actually believe. Consider that if children are the primary end of marriage, then the underlying purpose of marriage is to have and raise children. Thus, troubled marriages cannot, and will not, be abandoned because it exists, above all else, for the children. The Church is less likely to approve or acknowledge separation when children NEED their mother and father to grow optimally. Including a selfish end, like "mutual help" (which is vague and ill-defined), open up the possibility for separation (and then divorce and more likely re-marriage).
To the average Catholic, HV simply restates what the Church has ALWAYS taught...which it does not. The post-Vatican II Church upholds many similar changes, which are contrary to Tradition.
Another example is Nostrae Aetate, which elevates "ecumenism". Almighty God says, in Psalm 95:5 (96:5 when not using the Douay-Rheims), that "the gods of the gentiles are demons", and yet NA talks mostly about how wonderful non-Christian religions are, and how much truth they hold. Traditionally, Catholics have ALWAYS understood that ANY deviation from the Catholic faith will result in a complete and total loss of faith. The Catholic Faith is the one, true Faith. Everything else is false, and thus offensive to God. No pope before Vatican II would be caught getting a demonic "blessing" by shamans or non-Catholic "clergy", and yet Paul VI, John Paul II, Benedict XVI, and Francis have ALL embrace non-Catholic "clergy". In fact, JPII allowed a demonic idol of Buddha to be placed on the Tabernacle containing Our Lord at Assisi. Many Catholics who upheld the pre-VII faith viewed JPII's allowance as an act of apostasy.
Yet ANOTHER example: the use of the NAB and other modern translations of Scripture. The Church has, consistently and routinely, held that the Latin Vulgate is "free of all error". It is the norm of Scripture for ALL theologians...in theory. HOWEVER, at Mass, we hear the NAB, and most Bibles available to us, or used in our readings, are NOT based on the Vulgate. In fact, the NAB cites, in it's introductions to the Gospels, heretical statements condemned by Pius IX, Leo XIII, and Pius X, namely that the authors of the Gospels aren't known and that the Gospels themselves are derived from previous oral/written "traditions".
But WAIT...there's MORE! JPII said that theistic evolution was entire consistent with the Catholic faith. However, the IV Lateran Council, Leo XIII, and all the Church Fathers maintain that the Genesis account is the way it happened. Further, Our Lady called herself THE Immaculate Conception. If theistic evolution is correct, Our Lady was not THE Immaculate Conception, because Adam and Eve would have been immaculately conceived before she was. Either Our Lady and the Tradition of the Church is wrong, OR JPII (and the post-VII church by extension) was wrong.
I could go on, but that would take time. The bottom line, is that, when you actually look at the present state of the Church, and compare it with the pre-Vatican II Church, you see two distinct faiths. The problem is that the heretical Modernist group exists IN THE SAME SPACE as the True Church. In many cases, it is impossible to distinguish the two. Consider Bishop Wall's recent statement instituting ad orientam Mass in his cathedral. While it is laudable that he is doing so, there are some BIG Modernist assumptions that he makes in this (namely that error and truth can be equated, and personal feeling is a legitimate argument for celebration of Mass a certain way). He is assumed by most Catholics as a validly ordained bishop, a successor to the apostles. There is also the Declaration that Cardinal Burke and Bishop Schneider put out. While it DOES restate Church teaching, it includes post-VII accretions that are either highly suspect or definitely heterodox. We would normally consider Card. Burke "orthodox" or "conservative", but here he is, allowing these heterodox thoughts in some of his work.
One reaction to this is reject ALL of the post-Vatican II accretions, including the Mass and the Rites of Ordination, calling them "invalid". The only way THAT logically works is to state that the Council was invalid (as in NOT an ecumenical Council). The only way THAT works is if John XXIII wasn't actually pope. Of course, there are some who claim that evidence exists that John XXIII and Paul VI were Freemasons (and thus not Catholic, and so not pope because only Catholics can be pope). Others hold that both held heretical beliefs, and by default removed themselves from the papacy (which means that the New Rites are thus invalid, because they were put forward by an anti-pope). If the Council and the approving Popes aren't valid, then the rites of ordination aren't valid. IF this is true, than all priests and bishops ordained since 1968 are invalidly ordained, except those in the SSPX, as they never left the usage of the Traditional Rites. (Sedevacantists will argue that even the SSPX is heretical because it believes the NO to be valid--illicit, but valid--and deals with the "pretenders" in the Vatican as legitimate).
This is the REAL scandal of the Church: that heresy has so infected the Magesterium that even the Magesterium doesn't mention it. This means that the laity are in a bind. Those who followed St. Pius X's lead and went to fight Modernism see the errors, but have no one to turn to. Those who would like to promote Tradition and restore Holy Church are silenced by the Modernists who are in control. Groups like the SSPX felt compelled to violate the wished of the Vatican, to preserve Tradition, but yet recognize the Vatican as legitimate.
