An ombrellino looks like a fancy umbrella. In fact, it is. It is used to showcase what is underneath it. Hence it is used to be held over the Eucharist, carried in procession.
Basilicas have an ombrellino in one of their pulpits. It is closed, unless the pope is there. The basilica, IMHO, is just showing off that they're so important that the pope visits them. The real reason is that a basilica is the largest church in the area and hence big enough to hold the crowd.
