Ich habe diese Sätze gefunden :
Wo ist bitte das Cafe? - Das ist an dem Rosenplatz.
Wo ist der Markt? - Auf dem Kirchplatz.
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Basically, the difference is clear:
"An dem (oder am) Rosenplatz"
means that the Café is located adjacent to the place.
"Auf dem Platz"
means on top of it.
A "Markt" (market), not to be confused with "Supermarkt" (market center) in an European city, is comparable to a bazar, so it consists of "Stände" (counters with a tent) that are placed on the market place. See the image from a market in Germany (From de.wikipedia article "Markt"): .
Colloquially, sometimes the wrong preposition is used, but that is bad style.