Bronze is a lot heavier, denser than steel (which makes it a much better cannon barrel than iron, but not better than steel) but copper has always been more expensive than iron ore and with rare exceptionally pure deposits, requires steadily more sophisticated mining, crushing, separation through smelting and refining and then alloying with that not so common tin, zinc, or arsenic. Good steel also requires a lot of expertise and rarer alloys than just iron ore (hematite or magnetite, there's a lot of low quality iron ore out there too) that were unknown as alloys in medieval times, so you've got low carbon, silica-softened, low temperature smelting and then low temp heat treating...it's very difficult to make good steel then and then in small batches by master craftsmen who took their recipes to the grave.
I'd expect bronze actually was cheaper than decent steel then and only more expensive than iron work often called steel. Copper's a precious metal and in use for coinage as well as jewelry (bad sign for a metal you want to use a lot of and for mundane purposes.)
Cheap bronze would mean many more cannons, more geared wheels for mechanisms in saw mills, water mills, grain mills, etc. (bronze is far less affected by moisture than the quick to rust steel and iron of the time.)