At around 25min, When Winnie Eddington (Rebecca Hall) was serving an egg to Arthur at his desk, she is shown cutting the top off the egg (she even asks if she should "take the top off the egg") but the very next shot, the egg is clearly still whole and unbroken.
One the night before the total eclipse of the Sun, a full moon is shown in the sky. That is impossible since a solar eclipse can only occur at the time of a 'new Moon', i.e. when the Moon is the opposite to being full.
At the end of the movie, Einstein is shown getting his picture taken with his tongue sticking out. This photo actually happened much later in his life. On Einstein's 72nd birthday on March 14, 1951, UPI photographer Arthur Sasse was trying to persuade him to smile for the camera, but having smiled for photographers many times that day, Einstein stuck out his tongue instead.
During the partial phase of the eclipse of the Sun, the entire outline of the Moon was visible. This does not happen in reality.
The scene where Einstein insults the other University's scientists is supposed to be a day after the deployment of gas at the second battle of Ypres. Amongst the scientists is Fritz Haber. In actuality Haber would have not been present as he was present at the front to see the deployment of the chemicals.
In the Berlin café there are many German officers wearing hats or helmets indoors. German and Prussian military code forbade the wearing of hats or helmets inside, in fact in general it was not good gentlemanly manners to wear hat indoors in much of Europe.
Two Americanisations:
There was no rank of 'First Lieutenant' in the British army - it was just 'Lieutenant'
The Manifesto that Einstein refused to sign is shown as being 'To the Civilized World'. English spelling is 'Civilised'
In one scene, Einstein walks past a number of flags showing Germany's colors black, red and gold. This flag was not introduced until 1919. The flags at the beginning of WWI would have shown the colors black, white and red in horizontal stripes.
When Eddington went to the railway station to see Marston off, he used a subway to get across to platform 3. Cambridge station has no subway, all the platforms are accessible on the level.
(at around 27 mins) Boom mic seen in train scene in train station at background.