Firstly Seymour never actually ran, in both cases his name was put up at the party conventions in spite of his wishes.
But lets say you get him into office, it depends on what the circumstances are. In 1864 the only way you're getting him into the job is if the war is already over, with a Confederate Separation or a Union Victory that is as revolting as a Nazi Occupation.
In 1868 if you have him win the primary act of his term in office would be the fight between himself and Congress (unless it followed him into a Democratic win) to end Reconstruction in the South. The Democratic Party at the time is determined to turn the clock back to before the Civil War, so we'll see a very unpleasant abandonment of the Freedmen and Republican governments down there to the Redeemers.
Besides that, you'll never see talk of Annexations for Santo Domingo or Haiti, and Alaska won't be bought by the US, at least when he did it. Probably an economic problems based on his Treasury Secretary pulling out Greenbacks as quickly as possible.