It's a little bit more complicated than that, and a PBS article doesn't cover much.
Here is my understanding (sorry if it gets a bit complicated!).
There are two different types of immune responses. A humoral or antibody-based response, are fully developed antibodies that bind to antigens (specific proteins on viruses) and prevent them from replicating. Maternal antibodies fall into this category.
A cell-mediated response are killer T-cells that destroy infected cells. This is the kind of response induced by vaccination.
After infection (or vaccination), a certain number of killer T-cells become memory T-cells. When the virus starts replicating again and are encountered by memory cells, they produce a massive number of killer T-cells to fight the infection. This is what produces long term immunity.
These two types of immunities are also commonly known as passive and active immunity.
The level of maternal antibodies varies from chick to chick, and in general active immunity provides better protection than passive immunity. You want to give the chicks time to develop memory T-cells before actual infection, which is why it's so important to keep them from being exposed for 1-2 weeks after vaccination. That way when the chick is exposed, the memory T-cells can quickly generate killer T-cells and prevent symptoms.
Unfortunately, Marek's disease has the ability to become latent, which means some virus won't be seen by killer T-cells. This is why a chicken will never stop shedding the virus, even if it's vaccinated. But vaccination does aid in preventing symptoms and reducing the amount of virus a chicken does shed.
I hope this makes sense. I'll provide sources later.
