How important is the key of a song?

I’ve heard that the mood of a song can be completely altered just by changing the key. Is this true? I play the guitar and sometimes I’ll drop the key of a song down a few half steps just to make it a little easier to sing. Obviously there’s a noticeable difference when you hear both versions back to back, or when a key change happens in the middle of the song, but for the most part I can’t even tell. Does changing the key infringe upon the integrity of the songwriter?