This is not a post to argue for or against the big-O notation. Instead I just want to share an observation using a toy example.
Consider a dictionary data structure that supports access in logarithmic time. Suppose we have two implementations A and B where A is two times faster than B. (To make it precise but toyish, imagine an access takes $\log n$ time and $2 \log n$ time respectively where $n$ is the number of elements.)
How would this factor of two become important?
Well, when there is a hard upperbound on the amount of time allowed in an access, then the maximum number of elements that can be stored in A is actually the square of that of B.
It’s squared, not just doubled.
Now here’s something to think about. You can have a digital photo album that can store a maximum of two thousand photos or another one with four million photos. They are equally responsive on your computer. Which one would you prefer as a customer?
(Or, perhaps we can wait 18 months and be happy with the former one? 
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