Everywhere I look, every resource I read, says that time signatures determine the "feel" and meter of the music. And then the person giving the explanation will show a piece of music as an example of a type of some meter (usually a piece of music that doesn't have readily accessible sheet music); but then someone will argue that it's actually so-and-so a meter, and then a big internet argument ensues.
The time signature is supposed to determine the "feel." But I need hardly say that two different pieces with the same time signature can have very different "feels." A lot of Shostakovitch "feels" unmetered (or maybe I have no ear for music).
People will insist that certain beats will be stronger with certain time signatures. Yet I can find many examples in music where a different beat gets the accent, and also other theories of music that disagree on which beat should be accented.
The first movement of Beethoven's Fifth Piano Concerto is common time. Yet the grand piano solo at the beginning is shoved into one measure, regardless of the time signature, and nothing bad seems to happen.
I'm currently learning Beethoven's Eighth Piano Sonata, and till recently I misread it and thought it was in common time. Turns out it's not common time (4/4) but actually cut time (2/2). What's the difference? You still have four quarter notes per measure. How would it have felt different if it had been written in common time instead of cut?
And then you get Rachmaninoff, who will do weird things like change the time signature for one measure only. In Op. 23 No. 5, measure 16 changes to 2/4 time for one measure, and then back to common time. In Op. 32 No. 12, measure 13 changes to 6/8 time for one measure, and then back to 12/8 for the rest of the piece (I'd post pictures, but the website won't let me).
I cannot for the life of me figure out what purpose these time signature changes serve (and neither can my piano teacher).
The only purpose of measures I can think of is if a conductor wants to say: "All right, let's take it from measure 42!"