Reformed / Calvinist / Particular versus Arminian / General
The Reform movement began a Century before Calvin's time, but one of the debates that divide Protestants from Catholics and Reformed Protestants from other Protestants came into clear focus as a result of conflict between John Calvin and Jacob Arminius. For this reason, the Reformed school is often called Calvinist and the non-Reformed school is called Arminian. Within Baptist history, the Reformed position is held by Particular Baptists and the Arminian position is held by General baptists.
The differences break down into many points. In general, the Reformed beliefs are far more biblical than the Arminian beliefs. Here, I'm going to focus on just one point: For whom did Christ die? The Reformed belief holds that Christ died for the elect; that is, for, and only for, those whom God saves from damnation. The Arminian belief holds that Christ died for the world.
Both sides oversimplify by failing to separate the value of Christ's sacrifice and its application into two issues.
Reformed interprets "Christ died for the elect" too restrictivelyThe Reformed side errs by interpreting "Christ died for the elect" as meaning that the Sacrifice of Infinite God has finite potential value. First, this mathematical absurdity is reverse-engineered from the Catholic assumption that God's grace and sacrifice are quantifiable. Some argue that if all the value of Christ's sacrifice is not applied, then all is somehow wasted. However, dead is dead, and eternal God is infinite. It is absurd to quantify and limit the infinite.
Second, it is a logical absurdity to assume that having one purpose, dying for the elect, excludes having other purposes. For example, Christ's sacrifice glorifies the Father and Himself, so it would be correct to say that Christ died "for" glorifying God. By Reformed logic, Christ could not die for the glory of God or for vindication of the Father's plan to allow the Fall because He died (only) for the elect.
Third, Christ's sacrifice not only redeems the elect, but also, after the removal of the lost in the Day of Judgment, redeems the whole of creation. Yes, Christ died "for" the elect, but His sacrifice has far more value.
Arminian interprets "Christ did for the elect" too looselyThe other side understands that the Sacrifice of the Infinite has unlimited potential value but errs by confusing unlimited value with universal application. Arminianism holds that God applies the value to all men by freeing them to choose to receive or not receive Christ and His gift of salvation. This flies in the face of New Testament teachings about how the lost are enslaved by sin and how the carnal mind is at enmity with God. Paul makes it clear that even faith is a gift from God, and without that gift comes no conversion.
Many denominations, the Roman Catholic Church among them, carry the error further by falling into forms of Universalism, the belief that all men will be redeemed (or at least purged) and enter heaven. Pentecostals and Charismatics carry unlimited value to an absurd extreme by applying it to carnal desires for wealth ("it's in the atonement"), health ("by His stripes you were (physically) healed"), and personal pride ("he who speaks in tongues builds up himself").
On this point, both sides err by dragging along Catholic baggage. The error on the Reformed side does not affect any critical points of the gospel that I know of, but the errors on the Arminian side can be damning.
That is not to say that all Arminians are unsaved. One can believe the gospel before being taught Arminian errors, and not everybody who belongs to a given church believes all that their church teaches. However, I have only discussed one of many points. Those who have accept Arminian errors before conversion fall short of the grace of God. Their conversions are false, and those who mislead by preaching false gospels shall have much to answer for.