If a person with a mild to medium sprained ankle does no treatment but waits 2 weeks, will the result be the same as if he had iced & taken motrin?
Most likely. In most cases, a sprained ankle will heal no matter what you do or don't do. Ice and Motrin can help with the discomfort, but typically they do not change the outcome.
Likely no difference. Ankle sprains, especially low grade types, usually improve rapidly with or without formally therapy (rest, ice, compression, anti-inflammatory medication). Perhaps early aggressive treatment will hasten or speed up the recovery, but full recovery is expected, nonetheless.