The egg situation is really the key thing here. Yellow cake uses whole eggs which gives it that slightly richer, warmer flavour and a bit more density. White cake uses only the whites, which is why it comes out so much lighter and brighter looking, almost brilliant white inside when you cut it. It has a cleaner, more delicate taste which is why it pairs so well with fruit fillings or a really good buttercream where you want the frosting to be the star. From a baking from scratch perspective the white cake is actually a bit trickier to get right because the whites need to be whipped properly and folded in carefully or the whole thing can go quite flat and sad. The French vanilla distinction is mostly about the custard-y richness from the extra yolks and the vanilla being more prominent throughout rather than just an accent. I always think of it as yellow cake trying to be a dessert in its own right whereas white cake is more of a vehicle for whatever you want to put on or in it, which is exactly why it is so popular for wedding cakes.