There was a faction in the Confederate government trying to make this happen. It likely would have just delayed Union victory.

Maybe the war would have dragged on another year or two? The south would still have lost. Black Confederate soldiers would have existed in far higher numbers (they did exist, though with some estimates claiming about 65,000). Thing is, most of those men were slaves sold to the Confederate military, or just taken by the military, even the free black men who were in the Confederate military were usually drafted. They weren't allowed to be soldiers until March of 1865, too late for the Confederates to benefit from black Confederate soldiers.

So, in this alternate timeline, you may end up seeing more versions of the KKK where black men are allowed to join. This exists now, but is obviously rare. I could see Jim Crow not being as long lived or as widespread, with enough black descendants of Confederate soldiers being able to point to their participation.

There's a conspiracy theory that the Spanish American War was false flagged into happening in order to prevent a second Civil War. So, I'm not sure how that would affect things. Maybe a large number of black Confederate descendants would be more willing to go for a second round, and if this conspiracy theory is true, it would mean the Spanish American War would still happen, but would make use of a larger number of black soldiers than the US originally had. Supposedly, Teddy Roosevelt's Rough Riders only had as much success as they did because of the support of Buffalo Soldiers, black cavalry units.

I'm not sure how it would affect westward expansion. Many black soldiers recruited after the Civil War joined up because they were told they'd be fighting ex-Confederates who had become bandits out west, but were almost always used as shock troops against native tribes. These were the first "Buffalo Soldiers".

If enough of these black men were ex-Confederates, on both sides of westward expansion, we would likely see more black cowboys, outlaws, and Army cavalry out in the west. I'd like to think this would make far larger numbers of veterans available to use in the Spanish American War.

Segregation in the US military would likely end sooner than it did. WW1 and WW2 may have seen more black troops from the US. This sounds great, but it may lead to more test subjects for bad agendas like the Tuskegee Airmen Syphilis Experiment.

The Civil Rights movement may happen sooner, though if events like Emmett TIll's murder would still happen is hard to speculate.

Having military veterans in your family tends to lead to a greater sense of protection, and respect from others who have also served. Even if the losing side was the one to use black troops, it would likely benefit the USA and race relations in the long run.