In George Bernard Shaw's play Pygmalion, Shaw makes a distinction in Act 1 between St. Paul's church and St. Paul's cathedral. What is the difference?

Shaw refers
to the place where the opening of the play occurs as a portico of St. Paul's Church near Covent
Garden, where in the play, pedestrians take shelter from a sudden downpour of rain. Since Shaw
says "St. Paul's church" rather than simply "St. Paul's" or "St. Paul's
cathedral," this would distinguish it from the more famous structure.

St. Paul's Church, which was rebuilt in the 1790s after a fire burned down the original, does
have a very deep portico, and plays were sometimes performed there. From the point of view of a
realist play like this one, the portico is logically a place where people would
take...