It's actually a very important distinction. A vaccine just exposes a specific protein target to the host's immune system.

Thing is, each host immune system is going to create a completely different cocktail of antibodies to bind that protein. Suppose you and I got vaccinated for covid with the same vaccine. If you compared our resulting antibodies they would be completely different and also likely bind to different parts of the target protein.

The process of developing immunity is a pretty big subject and it's important to emphasize that.