What is the difference between the two?
The Society of St. Pius X (SSPX) is a traditionalist group founded by the late Archbishop Marcel Lefebvre. The group takes its name from Pope St. Pius X, an early twentieth-century pope who warned against the dangers of modernism. In 1988, against the wishes of Pope John Paul II whose permission was necessary to licitly ordain bishops, Abp. Lefebvre consecrated four bishops. The Pope quickly promulgated his response in his apostolic letter titled Ecclesia Dei:
In itself, this act [the unlawful episcopal ordination of four bishops by Abp. Marcel Lefebvre] was one of disobedience to the Roman Pontiff in a very grave matter and of supreme importance for the unity of the church, such as is the ordination of bishops whereby the apostolic succession is sacramentally perpetuated. Hence such disobedience – which implies in practice the rejection of the Roman primacy – constitutes a schismatic act. In performing such an act, notwithstanding the formal canonical warning sent to them by the Cardinal Prefect of the Congregation for Bishops on 17 June last, Mons. Lefebvre and the priests Bernard Fellay, Bernard Tissier de Mallerais, Richard Williamson and Alfonso de Galarreta, have incurred the grave penalty of excommunication envisaged by ecclesiastical law (ED 3).
In the present circumstances I wish especially to make an appeal both solemn and heartfelt, paternal and fraternal, to all those who until now have been linked in various ways to the movement of Archbishop Lefebvre, that they may fulfil the grave duty of remaining united to the Vicar of Christ in the unity of the Catholic Church, and of ceasing their support in any way for that movement. Everyone should be aware that formal adherence to the schism is a grave offence against God and carries the penalty of excommunication decreed by the Church’s law. (ED 5C).
Although in schism from the Church, the SSPX continues to recognize John Paul II as the reigning pontiff. On its web site, it affirms:
The Society of Saint Pius X professes filial devotion and loyalty to Pope John Paul II, the Successor of Saint Peter and the Vicar of Christ.
This profession of “devotion and loyalty” does not include obedience to the Pope’s authority to approve of episcopal ordinations (cf. Code of Canon Law 377.1, 378.2), but the SSPX does affirm that the Church has a valid pope.
The Society of St. Pius V (SSPV), on the other hand, believes that John Paul II is not the valid pope and that the papal chair is currently vacant (i.e., sedevacantism). The SSPV is a breakaway sect from the SSPX, founded when nine SSPX priests, disaffected by disagreement with the SSPX’s relationship with the “post-conciliar” Catholic Church, were expelled from Lefebvre’s group. Perhaps in reaction to the perception that the SSPX was overly friendly with the Church, the SSPV chose to take its name from Pope St. Pius V, the Dominican pope who promulgated the reforms of the Council of Trent in the sixteenth century.