Most of Western music is now played in 12et/edo, where a key change won't make an appreciable difference to how it sounds - providing voicings are the same.
Back in other temperamental days, let's take a piece written in, say, C major. If that was then transcribed to, say, Eb major, and replayed, would it actually sound any different, apart from being a m3 higher in pitch? Or, given the vagaries of instruments tuned to other than 12et, would it sound different?
It certainly would sound different. How different would depend on the particular temperament.
In the "mean-tone" temperaments that were widely used before J S Bach's time, and still used during his lifetime by some conservative organ builders, some keys were so out-of-tune as to be unusable. These temperaments were usually based on a cycle of 5ths from E flat up G sharp, and next lowest and highest fifths (i.e. A flat - E flat and G sharp - D sharp) were not usable, since Ab was a significantly different pitch from G sharp, etc.
Therefore, in your specific example, transposing a piece to E flat would probably sound horrible, since the A flats would actually sound as G sharps.
In the "well-temperaments" of Bach's time, all 12 major and minor keys were playable, but they certainly had their own distinctive sounds. If you study (and listen) to Bach's two and three part inventions, for example, you find that he makes deliberate use of the differences - and those pieces were written not only as teaching material for performers, but also for composers.
The same thing occurs in his major works.An example is the organ prelude and fugue in D Major BWV532, which starts with a passage of scales and arpeggios on a D chord, ending in a cadence on A. So far, all very nicely "in tune," but followed after a pause, by a lurch to F# major and its dominant C#7 - a "WTF" moment in unequal temperament, with the chords sounding as F#-Bb-C# and C#-F-G#-B! This is followed by another abrupt change, back to D major harmony.
This is a recording on an organ tuned to Valotti temperament where the tuning differences are fairly "mild" compared with some of the other baroque temperaments - but the effect is audible.
Contemporary writers noted that Bach sometimes used to "wind up" his favourite organ builder, Silbermann, who took a very conservative position on tuning systems, by deliberately using the out-of-tune chords when trying out a newly built instrument.
Modern digital organs (and other sound synthesis systems) often have the capability to switch between several historical temperaments. Experimenting with one will answer the question better than reading a wall of text!
It sounds different to me if people play in 12et or not, which makes different keys sound different. Of course, to what extent depends on how far the tuning differs from equal temperament.
