Is a brake flush the exact same thing as replacing the brake fluid?
Or is there a difference?
There is no difference between the two.
To replace it, you have to flush out the old brake fluid with new fluid.
See this answer on how to bleed brake lines for more details.
In some places, a brake flush might include some sort of additive used to clean the braking system. Its usually a gimmick.
Replacing the brake fluid itself is what it means. But flushes can vary between places and most of the time (from experience 99.999999999999%) its a gimmick to get your money.