'cents' are used 'as in a percentage of.' Since ears and therefore tuning are (mostly) logarithmic it makes sense (pun intended but accurate) to use this form.
Percentages, e.g. 4000 it to 2000 is like 400 to 200, would be the same amount of detuning, AKA 'octave' to tune one's instrument, or 50%. In case the first it was a difference of 2 thousand. In case the second it was 2 hundred. But the percentage remained the same.
Which begs the question; how can a difference of 2000 equal a difference of 200? Answer: that's why it's called logarithmic. and that's how people hear, how they see, and how they smell.
And while tuning to a small extent is individual, there are certain well recognized tempers that are accepted in Western music. (The scope of this comment isn't qualified to speak on 1/8th and 1/4 tones of Eastern and Middle eastern music.) Purely arithmetic tuning does not please the ear. Perhaps a "Well-Tempered Clavier" is the first pun in music history.
You have to 'fudge' some of the pure math to make pleasing chords when working with instruments that can span octaves.
I would suggest trusting in digital tuners, they have the tempering built in. But if nothing else, it's always better to be tuned to the band than be righteous about YOUR tuning. (p.s. Tune to the acoustic piano, they can't changes like git fiddles can.)
p.p.s. I feel so sorry for folks who actually have perfect pitch, (I worked with a bunch who claimed that, but NO!), nothing ever sounds right to them.