Fuzzyface is a name that has been unofficially given to an early version of Grover by fans. The name derives from a comment made by Kermit the Frog in a Season 1 sketch of Sesame Street. The term "fuzzy face" was actually a smug remark made by Kermit about the then-unnamed monster because he literally had a fuzzy face. In the same sketch, Kermit calls him "the hairy one" in a similar fashion (the monster also calls himself the hairy one).
The season one pre-cursor to Grover (previously seen on The Ed Sullivan Show) was performed by Frank Oz, just as the established Grover would be, but his appearance and mannerisms were somewhat different. His design consisted of a greenish-gray fur, and a slightly gruffer looking face. Oz also used a raspier voice for the character, and played him a bit more unkempt than Grover would later behave, certainly without any of the sweetness or consideration for others. Still, even in the first season, some characteristics of Grover could be seen, such as getting simple things wrong, being stomped on or exhausted, and slapping Kermit on the back. It was this early "proto"-Grover who first performed the well-known "Near and Far" sketch. Due to the differences between the season one Grover and the later Grover, who premiered in season two, fans applied the name "Fuzzyface" to his season one appearances based on the "fuzzy face" comment - creating an unofficial new persona designation for the character.
However, Sesame Workshop considers this green character to essentially have been Grover, despite not gaining the name until season two. In their 35th anniversary board game, one trivia question reveals that Grover was originally green, suggesting that these characters are the same, albeit models of the evolution of the character. Additionally, sesamestreet.org has "Old Grover" as a tag in the "First and Last" sketch from the first season (although the character has an orange nose in that sketch).
The puppet used for this "proto-Grover" appeared with the real Grover at the end of the "What Kind of Fool Am I" sketch on The Ed Sullivan Show in 1970 and appeared in a season 2 sketch, given a wig and a dress to become Grover's Mommy.