Peter Rabbit and Peter Cottontail are different critters.
According to the Wikepedia article, Peter Cottontail was “briefly” known as Peter Cottontail—thereafter as Peter Rabbit.
Actually, they're one and the same. Peter Rabbit, who was featured in many of the books Thornton Burgess wrote for young people changed his name to Peter Cottontail in The Adventures of Peter Cottontail (New York: Grosset & Dunlap, 1941).
Not too long ago, I read Burgess' Mrs. Peter Rabbit (New York: Grosset & Dunlap, 1919), which chronicles how the moody and depressed Peter sets out from his briar patch on a journey fraught with mortal dangers from predators as well as the jealous father of Miss Fuzzytail, a doe he is pursuing. Each chapter starts out with a quote from Peter, such as "Who has attentive ear and eye will learn a lot, if he but try.
Young people today ought to rediscover Peter Rabbit, a far better role model than, say, the bratty and boorish Junie B. Jones, popular among young readers today.