What's the difference between ''Specially'' and ''Especially''?

Notice that, they both are adverbs, and
Special    = Particularly purposed, in a distinguishing manner.
Especial  = Exception, in a noteworthy manner to a great extent.

in easier words
Special    = For special purpose [ just one thing is special for its own reason]
Especial  = To single out one thing [ comparison tend to involve here]

Check these examples for further help:

Incorrect:

Especially trained dogs were brought to the crime scene.

Correct:

Specially trained dogs were brought to the crime scene.

Incorrect:

Harry likes to watch sports, specially football.

Correct:

Harry likes to watch sports, especially football.

*Remember: in the sense of ''Particular'' both words would treat as the same.