For most people's ears, the difference between keys is just like you said. All of the notes in D are one whole step higher than the notes in C. Some of the more experienced, classically trained, musicians will go through the harmonic/overtone series and talk to you about tuning vs. tempering. I think I've met three people in my life who understand the concept and can hear the difference between the first and third note in C, and the first and third note in D. One of them was my Music Theory teacher in high school.

I have changed keys to many songs in order to accommodate my (or someone elses) limited vocal range. Also, if someone has a tin whistle (usually in D), everyone else is almost certainly limited to playing in either D or G (or the relative minors: Bm and Em).

Re-tuning the instruments is not really a workable solution because it takes too long. Also, with stringed instruments, you can only really tune the strings up or down so much before you start adversely affecting the sound of the instrument and possibly warping the neck. I'll use a capo before re-tuning my guitar.

Some songs seem to sound better in some keys than others, but I think it probably has more to do with open strings and the way people are used to hearing them than the miniscule difference between note distances within the various keys.