your thread title is in the other order, or it's two different questions?
A game for older firmware works fine on newer firmware.
A game for newer firmware needs to be patched to work on older firmware. Generally, decrypting/re-encrypting them, so you need the target's keys (4.4) to decrypt them, and your current version's key (3.55) for the re-encryption.
3.55 doesn't have 4.4 keys, so games encrypted with the 4.4 keys can't be read. the patch let you do it manually on your computer, using keys 3.55 knows and can decrypt. Usually, the patchers are re-encrypting with 3.41 keys to work on 3.41 and newer (3.55 is not the first hacked version)
Games patched to work on previous firmware than their original target should work fine after updating past your current version.
Some games can't be patched (Ni no kuni for example) because they require firmware files that were added or changed in newer version. changing the encryption will not be enough as you'll miss some functions required by the game.
So, updating is the only solution for these games.
I guess, 3.55 is fine as long as you get around the target version and patch all your games, but updating is easier (no need to patch anymore, and more games will work).
sometime the patch is not a simple re-encryption and needs file edition.