The actor playing young Stan clearly has brown eyes, but the elderly Stan has pale blue eyes.
Before he suffered his debilitating stroke, Oliver Hardy had lost 150 pounds, and was (for him) quite svelte at the time he had the stroke.
Stan Laurel tells Oliver Hardy about the Sputnik launch, which actually took place 2 months after Hardy's death.
At one point a document showing Stan Laurel's divorce proceeding is spun on to the screen. The two addresses visible have the United States Postal System's 5-digit ZIP Codes which were not created and made public until 1963. This is a British production which might explain the discrepancy.
Under the opening credits, an envelope addressed to Stan Laurel is shown that lists his zip code as [90210]-3218. The USPS didn't begin to use zip+4 codes until 1983.
A newspaper from 1935 includes a story about comedy team Martin and Lewis, who didn't meet until ten years later.