Different, probably. Incompatible, I hope not, otherwise all the great scientists throughout history and today who happen to be Christians are in quite a pickle. I’m not really sure what assumptions are involved, do you? I think people are probably pretty bad at assessing their own assumptions, it kind of defeats the purpose (you know, assuming).
I think science certainly has its own versions of revelation and faith, maybe that’s a starting point for compatibility.
Science seems to rely on the idea that nature has something to reveal. Of course in theology the revelation is different, but I think the central features of discovery, intelligibility, and reponse are found in both the scientific and Christian versions of revelation.
As to faith, as a grad student I sure had to have a lot of faith too. I had to have faith that if I followed the “rules of science”, I’d get something reliable out. That meant I had to have faith in my data, that it was correct and that I understood it. I had to have faith in my professors, that what they were teaching was correct and useful.
So, of course there are (significant) differences between “processes”, but there are also similarities which I think help keep each other in check and help support each other – which is why I call them “enriching”.
I’m not sure I could exactly list the axioms of science and Christian faith, at least explicit or intentional ones. I suppose for science I might say the intelligibility, coherence, and constancy of the universe might be a good start. Those don’t seem incompatible with Christianity at all, quite the contrary. I suppose for Christian faith the existence of God, his sovereignty and power, and his desire to reveal himself to humanity would maybe be examples. None of those seem incompatible with science, though they don’t seem to be necessary for science.
So I feel at least, that science is great, but it’s even better from a Christian perspective. I don’t think that takes anything away from science at all, but it should excite the Christian to learn more.