Alexander Hamilton is shown as part of Washington's staff during the Battle of Trenton. Hamilton was an artillery officer during the battle and would not become a part of Washington's staff until after the Battle of Princeton, several weeks later.
The British soldiers in the skirmish at the river bank are wearing uniforms from the French and Indian War, instead of the type in use during the Revolution. The cut is completely different, especially noticeable in the very large sized cuffs.
When Washington is lying on the bed the bottom of his boots are visible. The boots are curved for the left and right foot, but during the Revolutionary War, there was no such thing as a left boot or a right boot, they were straight.
At the end of the movie we find out what happened to the various characters. We are told that Gen. Hugh Mercer was killed by a gunshot wound, when in fact he was killed by bayonet wounds suffered in the battle of Princeton.
The weather, as portrayed, does not match the severe conditions during the actual battle. When the attack on Trenton began, it was a near whiteout blizzard - a factor which allowed the Revolutionaries to approach the town unobserved.
On December 20, 1776, outside Trenton, General Stirling tells General Washington that General Gates's army "will be with you by nightfall". This conversation is in full daylight, yet Washington then checks his watch and the time is 5:00 - by which time the sun would already have set. (And indeed, in a later scene we see it fully dark at 5:10 pm on December 25).