A vaccine causes the patient to develop specific immunity (fast reaction to a specific antigen) by introducing (usually injecting) a large amount of an antigen. That prompts the immune system to react by developing B-cells that produce their own antibodies to the antigen. It turns the body into its own antibody factory to deal with whatever you injected, but that takes time.
Injecting antibodies, as in this study, bypasses the natural (slow) specific immunity generation process. Antibodies are created some other way (in a lab, or in some other animal) and then directly injected. It's done for serious diseases or dangerous toxins or immuno-compromised individuals, when waiting for the patient to develop their own specific immunity is too slow or dangerous.
Critically, a vaccine allows the body to maintain the ability to fight off similar-enough infections for a while, but usually takes weeks or more to reach near full effect, sometimes requiring multiple shots for better effectiveness. If you're trying to treat an infection you think you might already have or will get in the next few days, vaccines only work if the progression of the disease is very slow, as in rabies which takes something like 1-2 months to make it into the CNS.
Injecting antibodies is a one-shot treatment that must be repeated until the virus (or toxin or other bad thing) is gone. It's commonly done for toxins (spider or snake venoms), or sometimes for severe diseases (ebola, tetanus), but sars-cov-2 has so much attention that now they're doing it for that too. In patients who aren't immunocompromised, antibody treatments for less-lethal infections give their immune systems more time to generate their own antibodies without developing severe symptoms.
The similarity in the two approaches is that they both ultimately involve antibodies binding to coronavirus proteins. The difference is in where those antibodies come from, and whether the body can produce more at will (vaccine --[time]--> natural immunity), or whether you have to keep injecting them (antibody treatments).
That's only a rough approximation. The immune system is very complicated. Wikipedia, or a book on molecular biology or immunology, will go into a lot more detail.