There definitely can be instrument-specific differences though, even under equal temperment. For example, the black keys on a piano sound distinctly different than the white keys; and on stringed instruments, some keys lend themselves to voicings containg more open strings, which AFAIK leads to a brighter/fuller sound (though I don't play stringed instruments, so don't quote me on that). There's also the practical matter that some things are easier or harder to play in some keys.

And of course, playing a song in a higher or lower key means you’re playing it higher or lower. A couple semitones can make a subtle difference in the character of a piece; transposing half an octave can be drastic. This combines with instrument-specific differences too, as most instruments just sound different in different octave ranges. I once had to play a piece written that was written originally in A major for a Rhodes electric piano, but I had to transpose to D major to accommodate a vocalist. This was incredibly awkward; transposing downwards sounded way too muddy, but transposing upwards sounded annoyingly chimey.