If Grant dies in Ford's theater, I don't it has any major impact. Lincoln surviving was covered in a recent thread.
Grant would have had a military escort. They all survive. Booth was in way over his head.
Even if Grant lives too, it's less certain that he'll be president without Johnson's interlude there. In any case, if Grant doesn't become president a few scandals might be avoided, though some would likely just affect some other president.
Grant's becoming president after the end of the American Civil War was considered a political inevitability. Horatio Seymour as the nominee of the Democratic Party of Treason was seen as merely a sacrificial goat.
If Grant dies in 1865, the butterflies are enormous.
The history of late-19th Century America is the history of people who grew up during and fought in the Civil War, now being handed the reigns of power for several decades. How this power was distributed between the 'big names' affected everything. In OTL, head chief was Grant, with his protégée Sherman, and hangers-on Sheridan and Schofield (plus many others). Whoever they wanted accorded political kindness was given it - whoever they wanted vilified was vilified (some examples including George Thomas, William Rosecrans, Gouvenor Warren, and George Meade to an extent). Besides whatever political and cultural power they had directly, they also dictated how the history of this era was written.
I agree about Warren and Rosecrans, but Thomas and Meade never came to political blows with Grant, even though he did not treat with them as he did his favorites. Both died before their respective careers could have gone far enough for them to be a serious problem for Grant. For Grant, Meade's performance in the Overland Campaign, the Siege of Petersburg, and the run to Appomattox was good enough for him, and his almost relieving Thomas just before the Battle of Nashville could have been a serious embarrassment for himself.
If Grant isn't around, then the first person who has a shot at filling his place is whoever was next-highest ranking - which is actually Thomas (having date of rank on Sherman),
Its Sherman who will get the Lieutenant Generalcy, as he had become a bigger war hero than Grant during his March to the Sea.
but he likely would not wield it. That said, Sherman only got his mythos in the public's eye because Grant promoted it, so I don't think he could fill Grant's shoes and direct this coterie.
Atlanta, March to the Sea, Savannah, campaigns through the Carolinas, taking Johnston's surrender of the entire remaining CSA military (tho it would take time for Kirby Smith in Texas to get the word)... Sherman didn't get where he was because Grant gave him a press treatment equal to General Douglas Macarthur's
What is interesting is the number of future political leaders, staring with Presidents James Garfield and Benjamin Harrison, who were Colonels/Generals in the war and who much more liked Thomas than they ever did either Grant or Sherman.
Garfield was never anything but the worst of political generals.
Also, while Sheridan (who would become General-in-Chief due to being in Grant's circle of friends) did like Grant best, he too was much closer to Thomas than Sherman.
Sheridan's fate is a tough call, but as the "Stuart of the East" from the Overland Campaign going forward, he'd have the role of public hero much more than the press hating Meade and the Virginian Thomas.
Also, the level of hate directed against Thomas by Southerners eventually got to him, leading to his early death in 1870.
Bottom line is, without Grant around calling the shots, the powerful ex-generals of the Civil War will re-group differently. Those of them that enter politics will have different relationships with each other, and promote different fellows. Scandals being avoided is just the tip of the iceberg for how things will play out differently.
Sherman abominated politics, and Sheridan followed in his footsteps, preferring to become general-in-chief one day after Sherman. TTL might see Sheridan make full general though, with another reaching lieutenant general.
But how likely is it anyway?
If Lincoln and Grant are both in that box, won't it be better guarded?
Yes
I remember in one book reading that as the Grants where on their way to leave town Mrs Grant Noticed they being followed her description matched Booth.
There are conspiracy theories even in as well understood an assassination as Lincoln's.
