Conciliarism is the doctrine that Ecumenical Councils bear authority over the whole Church, including the Pope, whereas Papal Supremacy, sometimes called Ultramontanism, holds that Ecumenical Councils are not supreme in themselves, but instead derive their authority from the Pope, who is over all.
The high point of Conciliarism came during a period of crisis for the Roman Catholic Church, when the Ecumenical Council of Constance (1414-1418 AD) was convoked in part to resolve the so-called Western Schism (1378-1417 AD) - a period in which there were multiple claimants to the Papacy.
By the time Pope John XXIII (1370-1419 AD) convoked the Council, there were two other Papal claimants - Pope Gregory XII (1327-1417 AD) and Pope Benedict XIII (1328-1423 AD). The result of the council was that John XXIII and Benedict XIII were declared Antipopes, Gregory XII resigned, and a new Pope, Pope Martin V (1369-1431 AD), was elected. Thus, an Ecumenical Council was required to heal the schism that had persisted for nearly 40 years:
Catholic Encyclopedia - Western Schism
After many conferences, projects, discussions (oftentimes violent), interventions of the civil powers, catastrophes of all kinds, the Council of Constance (1414) deposed the suspicious John XXIII, received the abdication of the gentle and timid Gregory XII, and finally dismissed the obstinate Benedict XIII. On 11 November, 1417, the assembly elected Odo Colonna, who took the name of Martin V. Thus ended the great schism of the West.
The Council of Constance also produced the most pro-Conciliar decree of any Ecumenical Council, formally stating that the Pope must submit to the rulings of Ecumenical Councils:
Council of Constance - Haec Sancta Synodus (1415 AD)
First it declares that, legitimately assembled in the holy Spirit, constituting a general council and representing the catholic church militant, it has power immediately from Christ; and that everyone of whatever state or dignity, even papal, is bound to obey it in those matters which pertain to the faith, the eradication of the said schism and the general reform of the said church of God in head and members.
Next, it declares that anyone of whatever condition, state or dignity, even papal, who contumaciously refuses to obey the past or future mandates, statutes, ordinances or precepts of this sacred council or of any other legitimately assembled general council, regarding the aforesaid things or matters pertaining to them, shall be subjected to well-deserved penance, unless he repents, and shall be duly punished, even by having recourse, if necessary, to other supports of the law.
Notice a few things about what the above states:
Significantly, the principle above - that an Ecumenical Council has power over the Papacy - was the means by which the schism was healed. The Pope who had convoked the Council, John XXIII, was at that time seen as the main claimant to the Papacy, whereas the one who would eventually be selected as the legitimate Pope, Gregory XII, was a rather fringe claimant in comparison:
Catholic Encyclopedia - Council of Constance
In the meantime (29 March, 1415) the English, German, and French nations had agreed to four articles, in the first two of which was expressed the complete supremacy of the council over the pope; these two were incorporated in the aforesaid articles of the fifth session. It has been maintained that these decrees were meant only for the extraordinary situation which then faced the council; they express, nevertheless, the well-known persuasion of the majority of the peculiar ecclesiastical representation at Constance that the council, independently of the pope, was the final depository of supreme ecclesiastical authority; indeed, by virtue of these decrees they proceeded at once to judge and depose John XXIII, hitherto for them the legitimate pope.
Catholic Encyclopedia - Council of Constance
The promised resignation of Gregory XII was now in order, and was accomplished with the dignity to be expected from the pope usually considered by Catholic historians the legitimate occupant of the See of Peter, though at this time his obedience had practically vanished, being confined to Rimini and a few German dioceses.
Notice that above, the Catholic Encyclopedia recognizes that it was "by virtue of these decrees", that is, the first two decrees of Haec Sancta quoted above, that the Council deposed Pope John XXIII, "hitherto for them the legitimate Pope".
Therefore, attempts to invalidate Haec Sancta run into the difficulty of the fact that without such a declaration, there would have been no means for the Council to resolve the schism. However, because the Council did resolve the schism, the tenant put forth in Haec Sancta proved itself - a Council fixed the Papacy - therefore, an Ecumenical Council must be supreme over the Papacy.
Following the Council of Constance was the similary-controversial Council of Basel. It was there that some bishops attempted to dogmatize the Conciliarism expressed by Haec Sancta.
This was denounced by Pope Eugene IV (1383-1447 AD), in a Papal Bull entitled Moyses Vir Dei (1439 AD), which would be incorporated into the Ecumenical Council of Florence:
Council of Florence - Session 7 (1439 AD)
The truth about the authority of a general council, representing the universal church, over a pope and anyone else whatsoever, declared by the general councils of Constance and this one of Basel, is a truth of the catholic faith. The truth that a pope cannot by any authority, without its consent, dissolve a general council representing the universal church, legitimately assembled for the reasons given in the above-mentioned truth or for any of them, or prorogue it to another time or transfer it from place to place, is a truth of the catholic faith. Anyone who persists in opposing the aforesaid truths is to be considered a heretic. (Condemned)
Notice that the Pope is acknowledging that the Council of Constance did in fact declare what he is denouncing, and those at the Council of Basel had attempted to do so as well. Yet, he denies an attempt to dogmatize it, on the basis that it was passed under dubious circumstances. Therefore, assuming that he rejects all the proposed "truths", and not only the latter ones, he is rejecting something proclaimed by an Ecumenical Council, which was used to heal a decades-long rift in his own office of the Papacy.
Likewise, by the time of the First and Second Vatican Councils, the Conciliar position lost completely in favor of the position that the Pope is supreme above all, and Ecumenical Councils are only authoritative insofar as they are accepted by the Pope:
Vatican 1 (1870 AD) - Chapter 3
8 Since the Roman pontiff, by the divine right of the apostolic primacy, governs the whole church, we likewise teach and declare that he is the supreme judge of the faithful, and that in all cases which fall under ecclesiastical jurisdiction recourse may be had to his judgment. The sentence of the apostolic see (than which there is no higher authority) is not subject to revision by anyone, nor may anyone lawfully pass judgment thereupon. And so they stray from the genuine path of truth who maintain that it is lawful to appeal from the judgments of the Roman pontiffs to an ecumenical council as if this were an authority superior to the Roman pontiff.
Vatican 2 (1965 AD) - Lumen Gentium
22 But the college or body of bishops has no authority unless it is understood together with the Roman Pontiff, the successor of Peter as its head. The pope's power of primacy over all, both pastors and faithful, remains whole and intact. In virtue of his office, that is as Vicar of Christ and pastor of the whole Church, the Roman Pontiff has full, supreme and universal power over the Church. And he is always free to exercise this power. The order of bishops, which succeeds to the college of apostles and gives this apostolic body continued existence, is also the subject of supreme and full power over the universal Church, provided we understand this body together with its head the Roman Pontiff and never without this head. This power can be exercised only with the consent of the Roman Pontiff.
The above statements are directly contradictory to the position expressed by Haec Sancta. If an Ecumenical Council is only authoritative insofar as the Pope accepts it, then Ecumenical Councils have no power whatsoever to reform the Papacy, nor is the Pope "bound to obey it in those matters which pertain to the faith", or else face punishment, as Haec Sancta decreed.
Therefore, the only tenable Catholic position is to reject Haec Sancta completely. It cannot be dismissed as only "disciplinary" - it established the point of doctrine that a Pope must submit to the rulings of "any other legitimately assembled general council", in addition to Constance. That is a doctrinal teaching on the relationship of the Papacy to an Ecumenical Council, not a matter of temporary discipline.
However, Haec Sancta cannot logically be dismissed completely. It was declared by an Ecumenical Council, and accepted as part of the acts of the Council. And, without using the principles that it expressed, the Council would have had no power to heal the Papacy, or depose the main Papal claimant, John XXIII.
The position of Papal Supremacy is repudiated within the Ecumenical Council of Constance, which is evidence that even to the 15th century and beyond (see the controversies that abounded in the proceedings of the First Vatican Council), those within Catholicism did not recognize the Pope as supreme over all, as today's Roman Catholic Church holds.