Emergency authorization and full authorization are not the same. One requires the recipients of the vaccine to be monitored for a longer period of time, as adverse impact might not be seen for months or even years.
And some people are old enough to remember the rushed into production swine flue vaccine of the 1970s that resulted in hundreds of people developing the paralyzing Guillain-Barré syndrome. I forget whether it was 60 Minutes or 20/20 that interviewed that poor woman who was perfectly healthy upon taking the vaccine, and was in a wheelchair and barely able to speak by the time they interviewed her for the story. So, some people might prefer to wait for full authorization.