PNG is the format to split when the pixels matter more than the file size.
It is lossless, so a tile written as PNG holds exactly the pixels that were in that part of the source. Split a PNG into nine tiles, put them back together, and you get the original file's pixels back with no drift. That is not true of JPG, where each round trip re-encodes and each re-encode changes something.
It also carries an alpha channel, which is the reason most PNG splits exist. Game sprites, UI assets, logos and cut-out artwork all rely on transparency, and any tool that quietly converts them to JPG replaces it with white. Tiles exported here stay PNG, and the transparent areas stay transparent.
What you pay for that is size. PNG compression works on runs of identical pixels, so it does very well on flat colour, type and interface screenshots, and quite badly on photographs. A photographic PNG can be several times the size of the same picture as a JPG. Nine tiles from a large photographic PNG can add up to a big ZIP.
A useful side effect: tiles are often smaller than a ninth of the source, because each one contains less variety than the whole picture did and compresses better on its own.
Processing runs on a canvas in this page. Nothing is uploaded.