Basically, "specially" means "for a special purpose".
Examples: "This suit was made specially for me." "The candles were bought specially for the party." "The house was specially built to resist fire."
"Especially" means "more than usual or more than others."
Examples: "The car was not especially expensive." "I like fruit, especially apples." "This is a small house to live in, especially if you have children."
However, nowadays these two words are often used the same way.
