Brass sponge has the mentioned advantage of not adding thermal stress to the tip when it cools, so it supposedly gives longer tip life. Other than that, it is mostly useful when you work with larger tips and need to remove large amounts of solder from the tip.
The downside of brass sponge is that it doesn't clean the tip 100%. There's almost always some solder remaining after using it. Which is normally a good thing, except when working with small/fine-pitch components (<0603 or <0.5mm pitch). If you have solder residue on the tip when doing such joints, it will mess everything up, so brass sponge isn't even an option for fine pitch work.
Also I find that tip life is mostly about using quality equipment with good temperature control, as well as avoiding polluting the tip. (Bumping into plastic surfaces of connectors, soldering onto lacquered surfaces etc. And stop using the solder iron for heating shrink tubes!) Covering the tip in solder when done working likely does far more for tip life than brass sponge vs wet sponge. I've had Metcal tips lasting over a year even though I exclusively used wet sponge.