(This question is aimed at the Roman Catholic and Eastern Orthodox perspective only)
What is the difference between a heretic and a schismatic, and what are the soteriological consequences of each?
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Heretic are they who restricting belief to certain points of Christ's doctrine selected and fashioned at pleasure. They believe not what Christ really taught, but the suggestions of his own mind. - (Summa Theologica - Second Part of the Second Part - Question 11)
Schismatics are they who of their own will and intention separate themselves from the unity of the Church. It is to be noted that Schism is NOT the same as disobedience to authority. Some disobedience can be schematic in nature. But not every disobedience is a schism; In order to become Schism, along with disobedience to authority, there should be denial of Divine right of the Authority to command. Its is to be noted that because of this in Catholic and Orthodox tradition all Schismatics are Heretics too.
Basil the Great explained the definitions of heresy and schism in his Letter to Amphilochius concerning the canons.
Heretics are defined as:
Men who were altogether broken off and alienated in matters relating to the actual faith.
Instances of heresy are those of the Manichæans, of the Valentinians, of the Marcionites, and of these Pepuzenes; for with them there comes in at once their disagreement concerning the actual faith in God.
Schismatics, on the other hand, are:
Men who had separated for some ecclesiastical reasons and questions capable of mutual solution.
Basil is considered an important Church Father by the Orthodox Church. His definitions are included in Protopresbyter Michael Pomazansky's Orthodox Dogmatic Theology, which is required reading at a number of English-speaking Orthodox seminaries and institutes (Fr. Michael was an Orthodox theologian and instructor at the Holy Trinity Orthodox Seminary in Jordanville, New York).
As Basil the Great is also considered a Father and Doctor of the Church by the Roman Catholic Church, I will propose that his definitions also apply for Roman Catholicism.
The EWTN (Catholic) expert Q&A contains this post on the subject:
The Catechism of the Catholic Church defines these three sins against the faith in this way:
2089 Incredulity is the neglect of revealed truth or the willful refusal to assent to it.
"Heresy is the obstinate post-baptismal denial of some truth which must be believed with divine and catholic faith, or it is likewise an obstinate doubt concerning the same;
apostasy is the total repudiation of the Christian faith;
schism is the refusal of submission to the Roman Pontiff or of communion with the members of the Church subject to him." [Code of Canon Law c.751]
The Church's moral theology has always distinguished between objective or material sin and formal sin. The person who holds something contrary to the Catholic faith is materially a heretic. They possess the matter of heresy, theological error. Thus, prior to the Second Vatican Council it was quite common to speak of non-Catholic Christians as heretics, since many of their doctrines are objectively contrary to Catholic teaching. This theological distinction remains true, though in keeping with the pastoral charity of the Council today we use the term heretic only to describe those who willingly embrace what they know to be contrary to revealed truth. Such persons are formally (in their conscience before God) guilty of heresy. Thus, the person who is objectively in heresy is not formally guilty of heresy if 1) their ignorance of the truth is due to their upbringing in a particular religious tradition (to which they may even be scrupulously faithful), and 2) they are not morally responsible for their ignorance of the truth. This is the principle of invincible ignorance, which Catholic theology has always recognized as excusing before God
Finally, the person who refuses submission to the Roman Pontiff, whom Vatican I defined as having a universal primacy of authority over the whole Church, is at least a material schismatic. It was thus common in the past to speak of the schismatic Orthodox Churches who broke with Rome in 1054. As with heresy, we no longer assume the moral culpability of those who belong to Churches in schism from Rome, and thus no long refer to them as schismatics.
Everything after the Code of Canon Law citation is not authoritative, in the sense that it is not an official document with the full weight of the magisterium, but it nonetheless addresses the soteriological consequences in a meaningful and Catholic way.
The Orthodox position on heresy is similar, though they more vocally reject Aquinas's work on the subject.
The Orthodox position on Schism is that it consists of cutting oneself off from the Church, and particularly in damaging the communion of the Patriarchs. With respect to the Pope, the Patriarch of Rome is believed to have been given due primacy by the Orthodox churches, but it is generally believed that, not content with primacy, the Pope demanded supremacy and that denying him that was not schismatic. More details on that particular issue can be found here
From Jaroslav Pelikan's 1st volume in The Christian Tradition (Ch2 3rd paragraph):
In its earliest Christian use, the term "heresy" was not sharply distinguished from "schism"; both referred to factiousness. But a dominant characteristic of such factiousness was that it created "dissensions and difficulties, in opposition to the doctrine which you have been taught." At least as early as Irenaeus, therefore, "heresy" came to be the term for a deviation from the standard of sound doctrine. It was consistent with this development that Augustine eventually came to define heretics as those who "in holding false opinions regarding God, do injury to the faith itself," as distinguished from schismatics, who "in wicked separations break off from brotherly charity, although they may believe just what we believe." Basil's distinction was only slightly different: heretics were "men who were altogether broken off and alienated in matters relating to the actual faith," and schismatics were "men who had separated fro some ecclesiastical reasons and questions capable of mutual solution." But already in the conflict with Montanism, even more in the conflict with Donatism, and above all in the church history of the West since the Reformation, the distinction between heresy and schism has not been easy to maintain with any consistency.