Aside from trying to reduce the effects of a warp, record weights and outer rings are attempts to change the sound to tune it to one's preference. It the weight more effectively couples the record to the platter/mat such that internal vibrations in the record itself are transferred more effectively to the platter and better damped (the usual intention of such weights) that MAY improve the sound, or it may make it subjectively worse; it really depends on what the rest of the system sounds like and what you are looking for in the sound.
I have Basis Debut table with a vacuum clamping system. This is pretty much the ultimate way to couple a record to the platter to dissipate vibration in the vinyl that will feed back to the stylus. If you place the stylus down on a record that is not spinning at tap the record surface with the back of a pen near the stylus, you can barely hear sound coming from the speakers. If you do that with other tables you will hear much more of that vibrational energy being fed to the cartridge. So, this is definitely a good thing? Not necessarily. Some people find this degree of damping to be excessive and subjectively "dead" sounding. It really is system and personal taste dependent. I've played my table with the vacuum off, and sometimes, I prefer the sound that way.