The switchless thing will use a microcontroller to change the region of the motherboard, you control this with the reset switch. This makes the console physically turn into a different region machine as far as software is concerned. The boot animation does not change, but if you have a non-Japanese machine, you'll see the different region codes in the Settings menu of the cd player.
AR will just patch this region code value on the fly, but you have to go through its boot menu when playing. It also blocks the cart slot and it cannot save in-game: you can only save from the AR menu. And the cart has a tendency to corrupt its own firmware - if this happens you can still re-flash it, but you'd lose your saved games in the process. The physical region code of the game stays the same however, so games checking the region after booting will play according to the region of your console.
Region-free bios knocks out the region detection from the boot rom, allowing all games to boot regardless of region. This is by far the comfiest solution as you don't need to mess with multiple resetting or going through an AR on every boot, games will just boot directly. The downside is that installing it is more difficult because you need to remove an SMD chip without damaging the board - on some revisions this chip is on the bottom of the console, and is glued down in a way that a heat gun won't be enough to remove it, so there's a chance you can damage the console if you are not careful.
The region-free bios also does not change the physical region code of the console, so games checking these will act according to the native region of the console. However I only know of two games to do this, one is Soukyugurentai which runs in half-broken English (a code remnant of the ST-V arcade versions localization), and I forgot what the other title was.
The upside to region-free bios is that you can change the boot animation by installing the bios of a different machine. Note that all bios versions are compatible with all Saturn hardware, just don't use the very first version of the Japanese, V-Saturn, or Hi-Saturn bios files, since those got a second revision with some bugfixes (they don't boot if the CD Drive is disconnected, while the later units will boot showing some errors).
I personally use a normal memory cart for in-game saving, plus a region free boot rom. Sometimes I change it out for the 4mb RAM cart, but truth is that most of those games aren't that much worth playing on the Saturn. In the 90s they were the best ports in existence, but by now the Saturn versions aren't that relevant anymore.