This is an engineering question. Any recommendation must be qualified pending additional data, as the OP provided insufficient information to support any blanket statement.
Is your local power dirty and/or does it suffer frequent voltage fluctuations? Does your LP-12 lack a regenerating power controller? If so, a conditioner or regenerator may be beneficial.
That dedicated circuit isolates your TT from other circuits in your house. It doesn't isolate it from the neighbors, the shopping mall across town or the crappy equipment at the local sub-station. In fact, dedicated circuits (and Stringreen's high end power cord) connect you even more closely to those problems.
* FWIW, my TT operates on battery power. It's only connected to the grid when I'm recharging the batteries. If I leave the recharging circuit connected when playing music there are audible side effects, even if the recharger itself is powered off. Noise from the grid is pervasive.
Conclusion: the more electrically isolated your TT, the better it should sound. The particulars of how best to achieve that vary with the individual situation.