Does anyone have an idea what this white fuzzy thing is?
No, that is not a sponge. I bought those mushrooms from my local LFS thinking that it was one large rock encrusted with corals, but instead it is a bunch of little rocks - each with a mushroom on it - and these are all held together with a green, hard, foam. So, that sponge is, in fact, foam. Thanks for all the other great information on sponges. I will definitely be googling 'sponge chemical warfare'.It looks like you have two different types of sponges there. As someone else has mentioned one is commonly referred to as a pineapple sponge. The picture is fuzzy, but it looks like you may have a yellow round sponge in front of the pineapple sponge as well. For the most part, sponges are harmless filter feeders, but they do get into chemical warfare with corals in your tank.