Big-O notation is there purely to compare one algorithm to another, yes. But in terms of what?
Say you have two search algorithms. For a dataset of size N, Algorithm 1 does N memory reads and N data compares. Algorithm 2 does N * (log N) memory reads and log N data compares. These are not real algorithms, just illustrations.
It's not uncommon in the analysis of such algorithms (both searching and sorting) to only consider the number of data compares and assume memory reads are completely free. So that would give you O(N) for algorithm 1 and O(log N) for algorithm 2.
Analyses with more sophistication will note that a memory read is not in fact free, and will look at the number of "operations", whether those be memory reads or compares. In terms of the number of "operations", algorithm 1 is O(N) and algorithm 2 is O(N log N).
These are both true statements about these algorithms: Algorithm 2 is O(log N) if you look at number of compares and O(N log N) if you look at number of "operations". Which of these characterizations is more relevant to you? Depends on whether memory reads are actually free in your setup.
Anyway, the original article is arguing that for the comparison most people care about, wall clock time, even the more sophisticated version is wrong, because it assumes that the time needed per operation does not depend on N, so you can just count all the operations and not worry about their relative speed, because that's just a constant factor. If the time taken for a memory read _does_ depend on N, then you can't do that. For example, if you want to look at the algorithmic complexity of these algorithms in terms of clock cycles, and a compare is one clock cycle (might not be, depending on what your data is!) and a memory read if sqrt(N) clock cycles (due to caching effects), then the complexity of algorithm 2 will be O(N * sqrt N * log N), whereas algorithm 1 is O(N * sqrt N).
Of course once you're caring about actual time very often constant factors start to matter too, and the whole idea of doing asymptotic analysis might break down. But it might not. It really depends on your exact problem and on the assumptions underlying the asymptotic analysis...