And if it makes sense - did ideological/philosophical concerns play a role in Roman persecution of Christianity?
I talked about the relationship between neo-platonism and Christianity in this post:
The religious aspect [of the appeal of neoplatonism] took on especial importance due to the relative harmony that could be established between certain neo-platonic ideas and Christian dogma. But, we mustn't forget that Plotinus, traditionally taken to be the virtual founder of the school, was not a Christian. While his Enneads had a profound effect on later Christian (and indeed Jewish and Islamic) thought, pagans as well as monotheists could be rightly characterized as neo-platonists.
That said, the pairing with Christianity does seem a natural one. One of the central upshots of neo-platonism is the underlying unity of some highest principle. Plotinus, at least in most English translations, refers to this principle as "the one". For a Christian, it is easy to identify this underlying, unified principle with God. Further, this underlying principle in some sense explains (or, to use more contemporary language, grounds) everything else. But, if this is so, then the neo-platonists assumed that this underlying principle must therefore be the most "real", in some sense. Again, there is a natural harmony with Christian dogma; the pure unity of the transcendent God (and, by extension, of the spirit) is a higher reality than the physical realm (including, importantly, the body).
But we should also note that while this seems like a natural pairing, it is by no means essential. The Stoics, for instance, could rightly claim to be at least as compatible, since they too shared a belief in the primacy of the "ruling center" above the body and affirmed the existence of an underlying principle that grounds all else. The writings of Seneca, for instance, were held in high regard by Christians.
Now, to get back to the question of the popular appeal... Abstract philosophical disciplines were typically the province of the well off, and much less popular with the masses than religion. Nevertheless, given the profound influence of St. Augustine, himself a neo-platonist or near enough, it is perhaps understandable that a certain amount of neo-platonist ideas would filter down to the masses via religion.
So, basically there is a link between platonism -- or more properly the neo-platonism that sprung up in late antiquity -- and the early church. However, I'm hesitant to make too much of this connection for two main reasons. First, neo-platonism was fully capable of standing on its own outside of any religious context, and it's influence was felt in pagan, Christian, Jewish, and Islamic thought. Second, Christianity survived and thrived long before being formally cast in roughly neo-platonic terms by folks like St. Augustine.
Moreover, Christian dogma, under the immense influence of St. Thomas, was later primarily characterized by a harmony between religious dogma and Aristotelian thought. It is this sort of theology that characterizes the bulk of Catholic dogma even today. So to claim that Christianity is simply "platonism for the masses" misses the mark by quite a bit, given that the theology of the largest and most influential variety of Christianity wound up more influenced by Aristotle than by Plato.
As an aside, I'll note that folks like Nietzsche (Russell is another great example) may have been first rate philosophers, but this does not mean they were first rate historians of philosophy. It's good to always keep in mind that the real aim of these sorts of folks is to make their own philosophical point, not to give you the most accurate reconstruction of the views of past philosophers. Best to take their comments on the history of philosophy with a grain of salt.
*Edit: Sources
More of a follow-up question, but wouldn't the differences between Christianity in ancient Rome and Christianity at the time of Nietzsche make such a comparison rather difficult?