Transfigurations are more like evolutions in Pokemon. For the most part the moves and stats gained on the mirage board of transfigurations are usable in all forms (ie if chocobo has cure unlocked on mirage board, you can still use it if you stay in chocochick form.)

The other option is more like unlocking a mirage in the same family that has its own set of transfigurations. Say you have two kinds of chocobo, regular chocobo and red chocobos. If you unlock the red chocochick mirage board, you could change the regular chocochick into the red chocochick. However, the red chocochick would be treated as a separate mirage and the abilities it knows will not be usable by regular chocochick. I don't know if there is a red chocobo in the game, I just couldn't think of a good example.

Basically both are good. Transfigurations make your mirage stronger and give you the option to make the mirage bigger or smaller. The other option let's you change into something different that will have different stats, abilities, strengths and weaknesses. Hopefully that made sense. It will make more sense as you play around with it though.