Is there a fundamental difference between the statistical methods of science, comparing medicine/biology with small sample sizes(n < 10^2 or 10^3) to the statistics applied in Quantum Mechanics (h: order 10^34) or statistical mechanics (N: order 10^23)
Well Statistics is a specific mathematics discipline, that like ALL of mathematics, we simply made up, with a set of rules governing the use of that particular discipline. That could consist of stated axioms taken to be valid, within that branch of math.
So the use of statistics, is independent of the "numbers" that you use its rules on. It matters not a jot, what those numbers are, the statistical rules remain the same. So physics or medicine apply the same rules. Those same rules can be applied to ANY set of numbers, no matter what their source. So you can apply the rules of statistics, to all the valid telephone numbers, in say the Manhattan telephone directory. It isn't any different from applying them to numbers obtained in a medical study, or a physics experiment.
Nothing in the rules of statistical mathematics, says the result is of any use or importance, for any purpose; that would depend on the origin of the numbers. The average telephone number, in the Manhattan phone book, can easily be calculated. It's of no earthly use, unless it happens to be your number. It might not even be a real phone number, but you can calculate it with the rules of statistical mathematics.
The utility of the stats results is entirely dependent on your purpose for doing it.
if you mean the statistical methods used to evaluate the significance in scientific papers. I should say there is no differences in methods, only the sample sizes and significance value can change depending on the subject.