Well, this topic is acctually deriving from urban traditions of XIX and mid XX centuries, however now retail managers tend to confuse names depending on better conotations in comsumer's consciousnesses on local markets.
Shopping gallery/galleria/gallerie is an interior inside shopping distict among other shops on high streets, but under roof. They are XIX century developments, which have turned attractive again, when cities started fight for consumer streams agains with retail industry and their shopping centers since late XX century (mainly in urban revitalisation in Europe).
Shopping mall/center is a detached building with shops inside in living or industrial areas in urban context but outside shopping area of the city (semi-competition to center of the city itsself) or located green field outside urban context. Mall is an Anglosaxon name for it, Center is continental European name,
The latter are mid XX century development originally for suburban residents as substitute for urban space. Later they were found intruding urban areas, competing with shopping areas of the cities, not only weakening the real estate market in city centers, but the liveability of the city and generating car traffic. Thus many cities started blocking new shopping malls/centers investments and reintruded investments in galleries to make shopping streets more attractive.