So mind you, I'm not an educated teacher but I've read pedagogy as a part of my museological training. You may thus want to go into more detail than I'd do.

I would keep it short and point out that the Holocaust included a systematic killing-industry with a clear end goal. After all, what made the Holocaust to the terrible event that it was, was not the targetted killing of human beings - such things had been seen before (both through exterminations of cities and catastrophes like the Black Death) - but that a full industry grew up just for that purpose. Not to mention The Final Solution.

If you then compare that to the persecution of Christians (or Jews for that matter) there are a few important differences to be found. The Christians were not sought out just because they were Christians, but because they caused trouble in the Empire. They were then targetted as a group in a sense, which is one of the similarities to the Holocaust that is obvious to see, but without the systematic killing or the industrial part. As far as I know (please correct me on this) the worst thing that happened to the Christians was the public slaughter at the amphiteatres.

Another important aspect that I'd bring up is the difference in justification. The Nazis justified the killing of jews by blaming them for the loss of WW1 (gays, gypsies - do we still use that term? - and others were of course also killed, but that's going into detail). The Romans on the other hand justified their killing by saying that the Christians caused unrest in the empire, especially then through their refusal to worship the emperor.

I wouldn't go into more detail than that myself. Obviously, if there is further interest after that then it's a different thing, but to answer that question I would stay at the surface with the pure extremeties.