What a BMP file actually is
BMP is Microsoft's original raster format and in normal use stores pixels with no compression at all: three bytes per pixel, plus a header. A 12-megapixel photo lands around 36 MB. An optional run-length mode exists but is rarely produced.
BMPs turn up from legacy Windows software, older scanner drivers, industrial and medical equipment, and Paint on an ancient machine. New files in this format are almost always a sign that something in the chain has not been updated in a long time.
What you get from PNG
PNG output is lossless, so the saved pixels are exactly the pixels that went in and re-saving costs nothing in quality. It keeps a full 8-bit alpha channel, which means soft shadows and anti-aliased edges survive against any background.
Universal since the early 2000s, and the format every design tool, browser and operating system handles without argument. It is also the safest format to hand to a printer or a colleague when you do not know their software.
What changes when you convert BMP to PNG
Both formats are lossless, so nothing is thrown away in either direction — this is a container change rather than a quality decision. What differs is how efficiently each one packs the same pixels, and how widely each is supported by the software you need to open it in.
There is no transparency to preserve — BMP files do not carry an alpha channel, so every pixel is opaque to begin with. Converting to a format that supports transparency does not create any; it only means you could add it later in an editor.
Expect the file size to collapse without giving anything up. Both formats are lossless, so this is the rare conversion with no trade-off at all: DEFLATE finds the repetition that BMP stores verbatim, and a screenshot or line-art bitmap commonly ends up under a tenth of its original size with identical pixels.