Furniture and Amenities are mainly the stuff that you could Craft in the game which in return provides you somewhat an expanded reach to materials that you initially don't have access to. Both the furniture and amenities are crafted by Cyrus (the game's crafts-go-to-guy). You can get to him by clicking icon between Items and the Map in the main tool bar in your UI. Furniture are mostly the items that the different animals are attracted. They are like pre-requisites to earn a visit from a certain animal, which in return, will give you access to a new material in the game. Other than that, furniture are also used to decorate your living space. Furniture that can be crafted in the game include chairs, tables, bookcases, etc. On the other hand, Amenities are the large structures placed behind your camp (by clicking the Tent Icon). Amenities can be built by opening the Crafting menu and clicking onto the tent like icon on the far right part of the tab. In comparison to furniture, amenities are far more costly to build, requiring a ton more resources and time to build.
What is the need to spend on Amenities? Amenities basically benefit you two ways. First, it will gain +5 friendship points with all the animals that are already in your camp site. This happens when you call out for a party to unveil your newly built structure (automatically happens in the game). That aside, the more important benefit of amenities is that they raise the max level of friendship with animals. Take note though that each type of animal (cool, cute, natural, sporty) have a specif amenity that they are likely attuned with -- similar to an elemental affinity of sort. By default (when you just begun playing the game), friendship level to animals can only reach up to level 7. Beyond that, every earned friendship point will have no effect whatsoever unless you have unlocked the appropriate amenity for each of the animal friend you may have.
Amenities in the game also usually have 3 stages to upgrade to -- to which each upgrade gradually increases the Cap Level of the affiliated animal/s. Take note that every upgrade you purchase becomes more costly than the previous one. That aside, completely unlocking the upgrade tree of each amenity (or the collection of amenities) could potentially earn you a friendship level cap of 20 per type of animal.