Schiller finished his play "The Thieves" in 1781, 4 years after Schubart's arrest, which he was aware of. In the movie however he finished his play around his arrest, which would be around 1777/1778.
The movie portrays Schiller as someone who was forced to study medicine. In real life it was advised by the Prince of Wurttemberg. He agreed to it in 1776 because he believed he could combine his love for poetry more with medicine than with law studies. Furthermore the movie shows him beginning his studies in 1777, not in 1776 as in real life.
In a scene set in 1777 Laura is reading the poem "The Triumph of Love" (Der Triumph der Liebe), that was released in 1782.
The poem "Die Größe der Welt" (The size of the world) was written in 1782, not in 1780/1, as in the movie.
Prince Karl Eugen von Württemberg mentioned that he doesn't have a son. While he didn't have official sons, as his only daughter Friederike Wilhelmine Augusta Luisa Charlotte died in 1751, one year after her birth, he had numerous illegitimate sons, including Friedrich Wilhelm von Franquemont, Eugen, Karl Borel, Karl David von Franquemont, Friedrich Graf von Franquemont, Karl von Ostheim and Karl Alexander von Ostheim who were all still alive at the time of the movie's setting of 1777-1782. He obviously didn't mention it as it would be inappropriate for a monarch to publicly announce it.
Schiller and his friends are quoting the Thieves song from his play "The Thieves", which was released in 1782. In the movie he just finished it (incorrectly set to 1777/1778) and since it was never published by that time it is very unlikely that the friends know these verses by heart, as in the movie.