(Photo credit to Christopher Marciano)
In practical terms, the new basilica, which is a home for the Pope when he visits a country, can now be regarded as a place of pilgrimage and is afforded special options for celebrating votive Masses. Special ceremonial items can be carried in procession and a formal cape can be worn by the officiant at major religious holidays.
As Duane L.C.M. Galles explains:
The traditional privileges of the minor basilica (besides the name) are: the right to display the papal coat of arms; the use in procession of a special red and yellow silk canopy which was once used to protect the pope from inclement weather during papal cavalcades; the right to use a bell mounted on a staff which in former times served both to marshall papal processions and to warn bystanders of their approach; and, if the minor basilica were a collegiate church, the right for the basilica's canons to wear, as choir dress in winter, a "cappa magna" of violet wool trimmed with an ermine cape, or in summer, a cotta or surplice, over the rochet.
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In addition, the building must be large and beautiful with appropriate furnishings, and enjoy a certain renown in the diocese. St. John's, founded in 1854 during a turbulent time in American history, certainly has this. Nonetheless, the designation is far from automatic. Pastor Rev. Msgr. Stephen M. DiGiovanni, H.E.D., submitted documentation from the entire history of the parish, extending to many hundreds of pages, and a full review was required.
The designation is particularly exciting for the cause of sacred music. The director here is Scott Turkington, who keeps an incredibly full schedule here at the parish doing precisely what the Vatican has long urged parishes to do. The parish is an exemplar in the use of chant and polyphony, and this designation sends a strong signal about the Vatican's wishes for other parishes as well.
The basilica can also display the conopeum, a large umbrella with Papal colors that was originally designed to protect the Pope from rain
Here is one of many images online.