aoneil,
Run Locally preprocess the document locally on the machine before sending it to the printer, while not checking this box lets the printer do all the job on it's own environment.
Generally, Run Locally is required when you use dynamic images from a computer's folder, or anything that requires local processing. "Not Run Locally"(renamed Printer Centric in newer releases. Run Locally is now Optimized PostScript) is often faster since the printer will only process this job and not run many things at one time like a computer does, although the difference isn't as big today as it used to be in the past, since machines are now much better. (which is why Optimized PostScript now has much more uses in v6, and even more coming for v7)
Also, due to both methods being different and not using the same type of processing, it is possible that a job might work better in one mode than another. This seems to be your case. If you need to use Run Locally, then we need to find out what causes it to not work properly. I would start by removing the image ressources if you have any, especially if they are pdf or eps, as they can sometimes interfere with form compilation due to them being PostScript images embedded within a PostScript document. Do not just remove the picture objects, remove them from the Image Ressources entirely, otherwise, they will still be processed. Once done, try to run it again in Run Locally, and see if it works better. In any cases, there is nothing wrong with not using it if you don't need it.
Regards,
Rapha