Cleanliness is important whether we are vaccinated or not.
Obviously hygiene matters, (short of wearing a completely sanitized bubble, though it's not really the strongest defense) but it's actually important that you come into contact with germs and dirt. Especially at a young age. With all our cleanliness and sanitation in the modern era (some people won't even allow their kids to play in the mud these days!) we've started to create "superbugs." This uber cleanliness approach actually does weaken the immune system.
See, an immune system is powerful, but it's not something born with all the information it needs, even if it's super health. In order for it to actually know what to do, it needs to come into contact with a virus. This gives it the information it needs in order to properly fight against it. So the next time it comes into contact with it, the immune system remembers it and annihilates it.
Having your kid eat a small amount of dirt, for example, is probably a lot more beneficial to the immune system than if you were to ban the child from doing so. Because it is giving information (albeit incomplete without a virus or simulation virus) to the immune system, which then stores it.
A vaccine is very similar in a way. It is not a real infection, but it mimics or simulates an infection (sometimes even with mild symptoms) which is considered a safer option than just getting infected by something so you have immunity next time (because some viruses leave you dead) and that information is then stored by the system so when it comes into contact with the same (or a very slightly different strain of the) virus it can just destroy it.
When an unvaccinated person gets a disease, it's the real thing straight up. Which makes the learning curve much steeper for the immune system. This means the immune system first has to figure out what the disease actually is, then it has to come up with a strategy to defeat it. This takes a lot longer if it hasn't had even a simulation, because it's essentially trying to figure everything out on the fly, without any previous "memories" or information if you prefer to help it. This makes the virus last longer in the body and it can then mutate (mutations are hard to predict) and it will be different by the time it gets to another person. And on and on the cycle goes until the virus is something completely different and we have to try and fight it all over again.