So Iāve tried to be as accurate in presenting their origins, please donāt shoot me!
(F)SSPX
The Priestly Society of St. Pius X was founded by Archbishop Marcel Lefebrve in 1970, originally with full permission and legal recognition of their local Bishop of Fribourg, Switzerland, Francois Charierre. The aim, and continued aim of the group was/is to preserve the Traditional Latin Mass and the formation of Holy Priests, thus that is why initially only ran a seminary, first at the University of Fribourg, then securing their own place in Econe, which still operates to this day. From the beginning they were critical of
CERTAIN aspects of Vatican II documents and there was much tension between the Society and French Episcopate, who refused to incardinate any of the priests who came out of their seminary. In 1974 the Vatican did a Apostolic Visitation, which reported favorably on the seminary and Society.
Yet, in 1975 the new Bishop of Friboug, Pierre Mamie, who wasnāt as sympathetic to the Society as his predecessor was, decided to withdraw his recognition of them, and they were asked to dissolve. This is where things start getting tricky and complicated. The Society refused to dissolve and continued to run the seminary, in 1976 Pope Paul reprimanded Archbishop Lefebrve, the Archbishop also decided to go ahead and ordain his seminarians, which the Holy Father also told him not to do, but the seminarians were ordained, and as a result the canonical penalties were that they were automatically suspended, including the Archbishop.
The situation remained as such for the next decaded the SSPX continued to ordain Priests, without canonical recognition, then, in 1987 Archbishop Lefebrve fearing that the end of his earthly life was near announced that he intended to elevate some Priests to the order of Episcopate to carry on his work. This of course was a problem in order to consecrate new Bishops, legitimately you need approval of the Pope, the then Cardinal Ratzinger, Prefect of the Congregation of the Doctrine of Faith, tried to work an agreement with the SSPX but it didnāt amount to much. Pope St. John Paull II tried to make personal appeal to Archbishop Lefebrve not to continue with the consecrations, but the Archbishop went ahead and on June 30, 1988, Archbishop Lefebrve along with a sympathetic Bishop Antonio de Castro Mayer consecrated 4 priests as Bishops. The two consecrating Bishops and the four who received the consecration were automatically excommunicated. Archbishop Lefebrve passed away in 1991.
In 2008 as a sign of goodwill the excommunication were lifted on the four Bishops, but no canonical recognition was given to the Society, which is now headed by Bishop Bernard Fellay. The stats according to them are, they have 6 seminaries, a memebership of 575 Priests, 3 Bishops( 1 of the original four was expelled in 2013),193 Seminarians, 102 Borthers. They are present on 6 continents, and organized into 15 districts.
SSPV
The Society of St. Pius V, I am not too familiar with, but what I know is that were founded in Oyster Bay, New York when they split with the SSPX in 1983 because they did not agree with using the 1962 typical edition of the Missale Romanum, but wanted to use older editions. While not explicitly sedevacantist, they are certainly sympathetic to that position and many priest from their ranks have been outspoken as being so. From the SSPV many other splits and independent groups have arisen, as far the Catholic Church is concern their leader Bishop Clarence Kelly is not a Catholic Bishop.
That sums it up very well.
