You can...

costumize the menu / interface in ways the company didn't implement themselves,

remove region lock to play games from outside your console're region ( and play the fan-translated patched version ).

play games without licenced cartridges ( either your own dumped games from your microSD card, or pirated versions ),

install mods / ROMhacks for your games ( legal but Nintendo dosen't want you to ),

install emulators with games not officially supported,

and as you said, 3rd party software.

On a jailbroken Switch, you can boot into the Atmosphere interface and do pretty-much whatever a non-jailed handheld console would do.

In the case of the 3DS, it means restoring online functions that Nintendo terminated ( though fan-made Pretendo ),

install costum themes and badges for the menu ( with Anemone and Theme Plaza ),

make a copy of your save files and then restore that save point ( Checkpoint ).

make StreetPass functions and MiiPlaza actually usable with Netpass

Install the 3DS port of the SNES9X emulator and play whatever SNES game you want ( or other emulators ).

If your volume button is broken, you can use the Rosalina menu to override the physical slider and adjust volume in software.

There is also a fan-run platform hosting the entire 3DS games catalog for free ( Nintendo no longer sell them ).

It's the only way to get 3DS eShop exclusives since Nintendo pulled the plug.