How much difference does the Key make?

When singing a song or playing a tune, how much difference to the performance does the Key you use make?

When songs and tunes were collected from Traditional performers at the time of Sharp, Gardiner, Hammond et al, the collectors wrote the tunes out in notation in particular keys.
We do not know that this was the EXACT key the source played or sung in, it could have been what the collector preferred, etc..
Also, it is possible that the singers (in particular) did not ALWAYS sing in the same key; it is less likely that musicians changed keys, i.e. did Harry Cox always perform songs in the same key?

Do we learn anything about the singer or the song by knowing the original Key?
Does it matter if it is changed?
Does changing the Key also change the song?

Tim Radford