How much difference?
The question implies, decidedly, there is some difference...let's safely say between 0 and 100% difference.
But, difference of what?
...ease of playing with other musicians?
...difference of enjoyment to the listener?
...difference of perception between the original composers's creation and the current performers' rendition?
I'm sure there are others unaccounted for...

Above responses eloquently hammer into each area, in non-defined response.

I say, if you want the listener to best be virtually transported to recall nostalgic circumstances reflective of their original exposure to a song - my personal belief is that it needs to be in the same key for most true "musicophiles", based on the following:
Recommended reading for everyone on the board "Musicophilia" by Oliver Sacks. (this is not spam, I have nothing to do with the publisher, Mr. Sacks, etc...)
The insight this book gave me into music, after already being recognized by my peers as a "musician", not merely a guitarist, made me feel as though I knew very little about music at all.
The Key of Clear Green: Sythesia and Music, chapter 14, brings forth an understanding on the topic rooted in the physical and nuerological effects of particular keys which I found fascinating.
...
when my band's vocalist says...I can't do this song, it's not in my key, there are 2 choices...don't play it, or change the key. Initially I try a change. At that point if it strikes me as inconsistent with the zeigeist of the song, if it makes my skin crawl or hairs stand on my arms, it gets canned. Sometimes it doesn't. Nothing worse than a Chipmunks version of a blues tune. If you play with your soul; listen with your whole body, not just your ears.