Factual error: Secretary of State John Hay is depicted issuing the order to "initiate hostilities" against Spain. Unfortunately, at the time we actually went to war with Spain, John Hay was not the Secretary of State. That position was held by William. R. Day. Hay did not become SofS until September of 1898, after most hostilities had ended.
Visible crew/equipment: While Teddy Roosevelt is gathering information on casualties in the first battle, a microphone pops into the shot at the lower left.
Continuity mistake: When the Spanish and German defenders of San Juan hill are loading their Maxim machine guns, the cartridge belt is being inserted from the left side. During the fighting, the cartridges are entering the chamber from the right, and the expended belt is exiting on the left.
Factual error: After capturing the gun emplacement, Roosevelt orders his men to "kill the Hun." That particular slang term for German soldiers hadn't yet been coined when the Spanish American War was being fought. It wasn't widely used until many years later, during the First World War.
Factual error: In Nash's voice-over at the end, he mentions that President Roosevelt is now in poor health. The subtitle has just identified the year as 1920. Theodore Roosevelt died in 1919.
Factual error: It's stated when Roosevelt arrives to take charge of the unit that he's new to military life. In fact, he was an infantry commander with the NY National Guard in the 1880s, so was no stranger to the military at all.
Factual error: When Hay is authorized by President McKinley to commence offensive operations, he hands the order to a Naval officer, saying, "Pass this along, Admiral." The officer in question is wearing the uniform of a Naval captain.