Your question is worded unclearly, but I think I get the general idea: why does the progression work? Why does the progression feel different if you change the chords?
I assume the progression repeats.
||: I vi iii V :||
One of the main reasons why the progression will work is because upon repeat you get the very strong defining progression V | I.
The changes happening between the tonic and dominant chords is all diatonic and subordinate in importance, in the bigger harmonic picture, than the tonic and dominant chords.
You could summarize the progression as ||: I ... V :||
If you swapped the two inner chords... ||: I iii vi V :|| ...it would change the feel and overall harmonic function that much, because it changes chords of subordinate importance.
...had a very different feel...if the chord underneath it is either going from vi-V, I-V, or IV-V...
It's hard to nail down the extent to which changes feel different. If you focus on fine details, surface levels, then any change will "feel" different, because it is actually different.
If you want to qualify difference, one way to do it is comparing how many pitches change between chord changes. There are three basic levels of triad changes: one pitch changes, two pitches change, all three pitches change.
When you have changes involving I, iii, and vi only one pitch changes from the I chord. So, by that measure I iii and I vi are roughly the same.
Let's say we choose I vi as the opening and then go to iii. The change from vi to iii is a change of two pitches. Alternately we could have gone to IV. The change from vi to IV is a change of only one pitch. In terms of number of chord pitches changed, those are not "equal" alternatives.
So, I vi iii and I iii vi are roughly "equal".
But, I vi iii and I vi IV are not "equal".
Again, equality here is only a matter of how many pitches change between chord changes.
Let's complete two of the progressions. We could have ||: I vi iii V :|| and ||: I vi IV V :||.
On a broad level the two are more or less the same being elaborations of ... V :||: I....
On a detailed level they are not the same, because ...vi iii... and vi IV... are not the same level of change of pitches between chords.
So, you get different "feels" on the detailed level of specific chord changes. They all "work" because on the broad level they both are functionally V | I.
