The earliest records of the family use the 'Rather' spelling. After they are in Tennessee, various branches seem to have changed the spelling to 'Wrather'. Sometimes, the same spelling occurs in the same county, so it doesn't seem to be only a variation by the writers of the records.
Family traditions in branches of both spellings seem to be that the 'other' side of the family changed the spelling. But, the earlier records never use the 'W' spelling.
In following the male descendants - they most all moved into Tennessee, from which they spread in all directions.
Two groups went up into Kentucky, one to the Allen County area - which used the 'R' spelling, and the other to the Calloway County area - which used the 'W' spelling.
Another group went into Alabama - they used the 'R' spelling (and they further moved into Mississippi & Texas).
Some stayed in Tennessee, and they used both spellings.
I'm not aware of any branch which went back to the 'R' spelling from the 'W', though.