It depends entirely upon the game. Games like Quake 3, thanks to incredible speed at which players would move, ensured that simply hitting the opponent was a feat. Headshots would simply result in certain weapons being far more powerful than intended. The railgun, for example, already delivered a solid 100 damage in a shot - enough to potentially kill a player instantly if they were without armor or any health boosts. Given that it was one of the easiest weapons to use at moderate or longer range already, such a change would result in the railgun being far more powerful while it's middle to close range counterpart (the rocket launcher) would be saddled with the inherent difficulty of applying a slow moving round to a target who's relative motion could be in excess of 40 MPH at ranges of less than 10 feet.
In today's modern slow paced shooters (by slow paced I mean that they restrict player motion to roughly what a human in peak physical condition could achieve), and the regular inclusion of hitscan weapons (weapons who's projectiles arrive in in the same frame in which they were fired) means that headshots do not unreasonably favor one weapon over another.