In Catholic countries*, certain acts that are fairly described as "religious" were civil crimes: publishing or preaching heresy, attempting to practice with craft, etc. Persons charged with these crimes were tried in canonical courts, because the civil judges lacked the expertise necessary to evaluate the charges, as well as the competence to determine orthodoxy. A canonical court would make its finding, ask the defendant if he wished to recant his position (if he was being tried for something like heresy that could be renounced), and enter a judgment accordingly.
The effect of the judgment was that the defendant was subject to penalties that were imposed by the law of the state and carried out by servants of the crown, not agents of the Church. So the Church didn't actually burn anyone at the stake, the state did. It was clear at the outset, of course, what the penalties for the crime were (these weren't always death).
The same was supposed to be true for interrogation techniques: clerics were prohibited from participating in torture. Did they sometimes? Almost undoubtedly, people break the laws in every age. But it was an actively reprobated abuse. I am inclined to say that torture of persons charged with canonical crimes was at some relatively early point prohibited altogether, but 1) I cannot recall the specifics and 2) because of secular involvement in these affairs the Church was not always able to enforce those rules.
I certainly think the distinction does make a difference. One, we don't go around ignoring jurisdictional lines when we discuss modern law. Nobody would say "I get my Medicare check from the State of Texas, and if I don't it doesn't make a difference because it's all the same." Furthermore, the imposition of the state's penalties, then as now, was fundamentally at the will of the crown: the monarch whose agents were carrying out penalties could, and surely from time to time did, commute or pardon persons convicted by ecclesiastical courts. (One might also note, as Austin did somewhere below, that although we find them crude today, the due process protections in ecclesiastical courts were far more advanced than those provided by civil systems until, in some respects, the 20th century.)
* England maintained a system of ecclesiastical courts long after Henry VIII's "problem," but I am not acquainted with whether they prosecuted religious crimes after that time or not. I think many of the recusant martyrs were tried by Star Chamber, although I recall some of them simply being tried at King's Bench. I know nothing of the judicial practice of Scandinavia and north Germany after the rise of Protestantism and very little of it before.