Good lab grade distilled is about the same as good lab grade DI. Depends on the quality and upkeep of your system. You need to check TOC/TDS/conductivity (or whatever standard best applies to your application) on the h2o coming out of your system, but both distilled & RO/DI are capable of producing very pure water. RO/DI isn't as efficient as far as water conservation goes, but distillation uses energy in the form of heat.
Grocery store stuff- well, I guess they're about the same as each other too. Often the water is stored in metal
tanks once it's distilled or ro/di'd. I've had problems with copper contamination in commercial distilled water, but that was years ago, and I dunno if that's a concern for your use.
Just make sure you avoid the de-ionized "with minerals added for taste" water from the store. Other than that, either is fine. Two different methods to get to the same end.