People who have any of the other respiratory viruses (common cold, flu, SARS-CoV-1, MERS, etc) are only infectious to others when they feel sick (symptomatic). SARS-CoV-2 is unique in that people who have this virus are infectious to others before they feel sick (pre-symptomatic) and even if they never feel sick (asymptomatic). I've seen published studies which have estimated up to 40% of all SARS-CoV-2 infections are asymptomatic.
Clinical trials for vaccines follow 10,000s of vaccinated (with vaccine or placebo) people for 2 years. During this time the effectiveness of the vaccine is evaluated by determining how many vaccinated people became infected vs how many who received the placebo became infected. With the other respiratory viruses it is easy to know if someone is infected because they feel sick (symptomatic). With SARS-CoV-2 how do you determine everyone who becomes infected if some infections never cause symptoms? How do you determine the effectiveness of a vaccine to prevent all SARS-CoV-2 infections (asymptomatic and symptomatic)? The SARS-CoV-2 vaccines are only evaluated at preventing symptomatic infections (the ones that can be identified), hospitalizations, and deaths. The vaccines are very effective at preventing hospitalizations and deaths.
A vaccinated person could become infected (asymptomatic) and then infect another vaccinated person leading to a mild (symptomatic) or asymptomatic infection A vaccinated person is unlikely to become sick enough to require hospitalization or die). A vaccinated person could infect a non-vaccinated person leading to asymptotic infection, mild symptomatic infection, hospitalization, or death. This is why vaccinated people should still wear masks.