Sergalpony 
I actually just had this discussion with someone, lemme drop some knowledge here!
First, the definition!
anthropomorphism
the attribution of human characteristics or behavior to a god, animal, or object.
Furry:
an enthusiast for animal characters with human characteristics, in particular a person who dresses up in costume as such a character or uses one as an avatar online.
Keep in mind, Sapience is not exclusive to humanity, whether you believe in aliens or not, it is a statistical impossible that humanity can be the only sapient creature in the universe, calling every creature with sapience anthropomorphized would actually probably be extremely offensive if it were done in a period where multiple species held sapience, as it implies they're all just 'like humans'
This is why there were arguments once upon a time between furries being inherently into beastiality [which is illegal in reality in various places for various reasons] because beastiality requires a 'feral' mind be involved, while it's just more 'common' among furries to be into that, it's not an inherent characteristic of a furry.
In a similar vein, being into 'sapient quadrupeds' is not an inherently furry characteristics, there-by a fandom centered almost exclusively around a species, or world where almost all species, are sapient quadrupeds, is kindof reductive, and the same line of thinking as those people who claim there's no biological difference between male and female.
And that is 'all' 'before' considering that ponies are so wildly different from any animal that we know of merely being 'anthromorphiszed', that they are more like aliens first, than just an animal being anthrofied.
Furry is for people into anthropomorphized animals, not 'everything with any human characteristics'.
While Brony is for people specifically into species from MLP, the vast majority of which share only 'sapience' with humans, which is not an 'exclusively' human characteristic.
Basically, this is a very long winded and in-depth way of saying, you can be a furry, and/or a brony [or whatever term you prefer], but they are not mutually exclusive nor identical! XD
As Ponies are not an 'animal that has been given human characteristics' so much as a naturally sapient quadrupedal species, they are not a definitive 'furry' thing, they're actually a sapient species that would need to be given 'more' human characteristics to be considered anthropomorphized to fit the qualifications.
First, the definition!
anthropomorphism
the attribution of human characteristics or behavior to a god, animal, or object.
Furry:
an enthusiast for animal characters with human characteristics, in particular a person who dresses up in costume as such a character or uses one as an avatar online.
Keep in mind, Sapience is not exclusive to humanity, whether you believe in aliens or not, it is a statistical impossible that humanity can be the only sapient creature in the universe, calling every creature with sapience anthropomorphized would actually probably be extremely offensive if it were done in a period where multiple species held sapience, as it implies they're all just 'like humans'
This is why there were arguments once upon a time between furries being inherently into beastiality [which is illegal in reality in various places for various reasons] because beastiality requires a 'feral' mind be involved, while it's just more 'common' among furries to be into that, it's not an inherent characteristic of a furry.
In a similar vein, being into 'sapient quadrupeds' is not an inherently furry characteristics, there-by a fandom centered almost exclusively around a species, or world where almost all species, are sapient quadrupeds, is kindof reductive, and the same line of thinking as those people who claim there's no biological difference between male and female.
And that is 'all' 'before' considering that ponies are so wildly different from any animal that we know of merely being 'anthromorphi
Furry is for people into anthropomorphized animals, not 'everything with any human characteristics'.
While Brony is for people specifically into species from MLP, the vast majority of which share only 'sapience' with humans, which is not an 'exclusively' human characteristic.
Basically, this is a very long winded and in-depth way of saying, you can be a furry, and/or a brony [or whatever term you prefer], but they are not mutually exclusive nor identical! XD
As Ponies are not an 'animal that has been given human characteristics' so much as a naturally sapient quadrupedal species, they are not a definitive 'furry' thing, they're actually a sapient species that would need to be given 'more' human characteristics to be considered anthropomorphized to fit the qualifications.