In the scenario you describe, comparing two services from the same non-specialist provider, there is no difference. They only have one set of equipment, and it's used for everything. The difference is that someone has to stop what they're doing to handle the one-hour printing, while the "regular" service can be done at their convenience.
In the days when film was king, there was a difference. "One-hour processing", which often took several hours, was done in-house right where you dropped off the film. The same shop would usually also offer standard service that would take about 5 days, because it was sent out to a processor.
The one-hour specialists tended not to change the chemicals used as often as they should have, and some failed to clean the equipment thoroughly. The slower services were generally performed by large-volume processors that did nothing but develop and print film and therefore relied on their reputations for their success.
One-hour shops tended to rely more on the tourist trade, and weren't as worried about repeat business, so the slower service could result in better negatives and, theoretically at least, better prints. But only experience could tell you for sure. Some one-hour services were excellent; some not so much.
Some of those large film processors still survive, and a number of smaller one have sprung up recently due to the resurgence of interest in film photography. Generally speaking, you'll get excellent results from any of them these days, but one-hour service is no longer an option for film shooters.