Especially is used to say that something applies more to one thing than to others.

Specially is used to say that something is done for a particular purpose.

Before adjectives it's better to use especially to emphasize them. For example, especially good (but not specially good).

Especially cannot be used at the beginning of a sentence.
She loves vegetables. She especially likes carrots (but not Especially she likes carrots).
especially
We should practice more, especially grammar.
Everybody likes this cake, especially the children.
This part of the film is especially interesting.


specially
This mouse is specially designed for left-handed people.
I made this cake specially for you.
She specially liked the cake.