I believe the major disadvantage is that there's no clear evidence that all of the potential union members have had an opportunity to seriously consider the question of unionization and to express their opinion behind the screen of a ballot box in a secret ballot vote. When you think about unionization, you could face a situation where in 1950, the employees at a certain work site signed cards to demonstrate that 50% plus one of the members joined a union. But you would never have any evidence about what the real wishes were of the employees who were not approached, did not participate in the process, and did not sign cards. This is the reason why employers would favour a secret ballot vote. It's just a better arrangement to determine the true wishes of the individuals who will be affected by the decision to unionize or not, that they have an opportunity to express their views.
On February 11th, 2014. See this statement in context.