Nope the difference is in both the alloy and the flux. Plumbers use acid flux stuff that will rot your wires. For electrical work you want either 60/40 tin lead or even better the eutectic 63/37 alloy.
Flux used for electrical stuff is basically pine tree sap which is acidic when hot but neutral when cooled to room temp. Still I'll usually clean any flux residue off with a little rubbing alcohol, mostly because I'm a neat freak.
You're less likely to produce bad or what's called a cold solder joint with the eutectic stuff. You can get by with the more common and cheaper 60/40 stuff as long as you're careful to keep the joint still and un moving until the solder cools and solidifies.
Plumbers solder is either 50/50 alloy, tin costs more than lead, or either lead free these days since it's likely to be used for drinking water pipes.