If I understand correctly, in this case you can directly compare them - the algorithm is the same, so the constant factors are the same.

However, this is in so many ways not my field, so I could easily have completely misunderstood.

You do not understand correctly. Compare the ratio of the quadratic f(n) = 1E100 + n2 to the linear g(n) = 1E100 + n. Using n=100 and n=1000 the ratio is effectively 1.0. You need to get to n=100000000000000000000000000000000000000000000000000 before the ratio is 2.

I see your point - we can compare the powers that n is raised to, but without knowing the constant factor we can't know the proportion of the result that it is responsible for.