Each and every viewing application has its own font rendering engine.
It has less to do with "web" and "desktop" than it does with the applications (including web browsers) themselves.
The font rendering engine in Edge is not the same as the font rendering engine in Firefox, which is not the same as the font rendering engine in LibreOffice, which is not the same as the font rendering engine in Adobe applications, which is not the same as the font rendering engine in Chrome... etc...
Each and every application dictate what to read from a font file and how to draw that font. The fonts themselves don't control display... only the data for the font. How that data is interpreted is up to the application developers. If the font has internal suggestions about how to draw glyphs (hinting) or metrix for kerning, etc that does not mean any application must read and honor those suggestions.