The difference is that an originating court is the court in which the trial is first held while an appeals court is one that hears any disputes on legal issues that came up in the course of the trial.
An appeals court will typically only hear appeals on issues of law. That means that the party that loses in the original trial does not usually ask an appeals court to rehear the case simply because they think the jury came to the wrong verdict. Instead, they typically only appeal if they feel that the judge in the original case made a legal mistake.
Pohnpei has a good explanation, one that I want to add a bit to. An originating court is often referred to as a "trial court," because, as has been pointed out, there is a trial, with witnesses and evidence. A court reporter is present to record everything that is said. If a case is appealed, the court reporter prepares a transcript. That transcript and all the exhibits entered at trial become what is then referred to as "the record." When appellate judges hear a case, they see only that record. They hear arguments from opposing counsel, but there are no witnesses with new testimony, and no new exhibits are introduced. The lawyers and the court are bound by the record, which means that the decision is based only on what was presented at the trial court level. This is why, in movies and on television, we see attorneys making objections "for the record." It is only that record that they have to make arguments to an appellate court.