The following was originally posted as an answer on Quora to the question at hand.
First, unlike the Roman Catholic Church, Presbyterians are what is called a confessional church, meaning that they adhere to a statement of faith (or “confession”) beyond the basic Creeds of the early Church (Nicene and Apostolic). A confession lists a whole series of things their members are required to believe. Usually for Presbyterians this is means the Reformed confessions like the Westminster Confession (see Westminster Confession of Faith). Most Protestant denominations fit this description — Lutherans, for example, adhere to the Augsburg Confession. A list of Reformed confessions can be found at Historic Church Documents at Reformed.org. The Roman Catholic Church does have its dogma and catechism, but no binding precise statements of faith beyond the Creeds. Technically when one becomes Catholic and is baptized, one need only accept the Nicene and Apostolic Creeds as the minimum statement of faith; the rest is therefore less binding, if still important.
Second, as the name implies, Presbyterians do not believe in episcopal governance like Roman Catholics, Anglicans/Episcopalians, Old Catholics, or the Orthodox do (in theory Methodists also fit this category, but are somewhere in between the two). I would call these latter churches collectively “Catholic churches”, even if some are technically also Protestant. The Catholic churches all believe in the three-fold offices of bishop, priest (also called “presbyter” or “elder”), and deacon. Presbyterians, by contrast, do not believe in an office of bishop that is distinct from the office of presbyter.
Third, as an extension of the second point, Presbyterians do not believe in the apostolic succession and historic episcopate like the Catholic churches do. This means that for them, only Scripture and correct teachings give a church legitimacy (Sola scriptura); unlike Catholics they do not require that the church be led by leaders in an unbroken line of bishops going back to the Apostles. The Catholic churches instead believe that Scripture is a source of doctrine, but not the only one — writings of the Church Fathers and other parts of Church tradition are just as important.
Fourth, as a further extension of the second and third points, they do not believe in an ontological change by ordination. This means that they don’t view ordination as a full sacrament like the Eucharist or baptism. A sacrament like baptism is irreversible — once baptized, always baptized. Thus once someone becomes a Catholic priest or bishop, they remain so for the rest of their lives, even if they may lapse in the meantime or stop actively working as such. An elder in the Presbyterian church, however, merely holds an office and is not “changed” by the ordination.
Fifth, there are major differences in teachings revolving around justification and free will. As a church in the Reformed tradition, Presbyterians hold to a kind of predestination, which Roman Catholics reject, preferring the idea that human beings are capable of using their free will to turn to God (while being unable to do so without help from the Holy Spirit). They also believe that faith alone (Sola fide) brings salvation, that is, only what one believes is relevant and not what one does. Catholics generally hold to faith and works, with the works (things you do) being the necessary result of that faith.
Lastly, like other Reformed churches, Presbyterians accept only two sacraments, Baptism and Eucharist, whereas Rome believes in seven (Baptism, Eucharist, Confirmation, Holy Orders/Ordination, Confession/Reconciliation, Matrimony, Unction/Anointing of the Sick). Furthermore, Presbyterians reject any semblance of the Real Presence in the Eucharist. The Real Presence means that one believes the bread and wine are somehow changed into the Body and Blood of Christ. Roman Catholics believe in a form of the Real Presence called transubstantiation, which was a medieval attempt to philosophically explain the Real Presence using Greek substance theory. Other Catholic churches generally believe in the Real Presence, but reject Rome’s narrower definition of it. Presbyterians view the Eucharist (or “Lord’s Supper”, as they prefer to call it) as symbolic only with no change in bread and wine. Furthermore, Presbyterians reject the Catholic idea of the Eucharist or Mass as a kind of penitential sacrifice, claiming that Jesus’ sacrifice on the Cross was a once-only thing for all time and that therefore “repeating” it at the Mass is heresy. (Catholics typically respond that they do not repeat that sacrifice, merely make it present in the here and now.)
