The Points of Departure are that Emperor Napoleon III of France throws his full support behind the cause of the Union during the American Civil War, and also does not get deeply involved in Mexico, withdrawing with the rest of the Treaty of London Powers in mid-1862. The French defeat at the Battle of Puebla leads to an early and drastic reform of the French Army.
Napoleon is able to persuade the British government also to withdraw it's grant of belligerent rights to the Confederacy, cutting the C.S.A. off from British arsenals and shipyards, as well as those of France. The Civil War ends in April 1863, with a complete Union victory and a formal surrender by the Confederacy. Because the Union is never in danger of losing the war, the Emancipation Proclamation is never issued. This leads to a very different Reconstruction after the war.
Because of Napoleon III's policies during the Civil War, relations between the U.S. and France are much better, post-war, than in OTL. When the Franco-War breaks out, America overtly supports the French by sending huge quantities of arms and munitions. The Prussians counter this by sending out commerce raiders which prey on American shipping. This leads to a Declaration of War by the United States. Hundreds of thousands of Americans…commanded by famous generals from both sides of the Civil War…go to France to fight the Germans.