Once again you are confusing GOP with conservative.
There is a difference.
There is a difference but not so great as after both made their Faustian bargain with the religious right and social conservatives, the very faction that opposed the Civil Rights movement.
Indeed, the success of the Civil Rights movement drove social conservatives from the Democratic Party to the republican during the late 60s and early 70s. Republicans, desperate to break the stranglehold democrats had on Congress, accepted these radicals and extremists believing they could control them.
The rest is infamy