Larry Mondello is the Devil
Think about it. He always shows up with apples, frequently tempts Beaver to do something wrong, and whenever Beaver gives in, he typically gives him a guilt-trip about it, even though he was the one who goaded him into it in the first place.- Also, given the general dopiness of the character, it's not hard to guess which portrayal of the devil the show was going for with him.
June Cleaver eventually became Nanny in Muppet Babies (1984)
This explains why they sound alike. Some time after Still the Beaver and The New Leave It to Beaver, the widowed June decided to start fostering children whose families couldn't take care of them. She ended up taking care of eight very unusual toddlers.