In many cases, there isn’t much difference between using “especially” or “specially”. Roget’s Thesaurus gives each word as a synonym of the other, and suggests “in particular” or “specifically” as further synonyms.
There is a small difference – when you use “especially”, you are usually singling out one particular person that you have done something for, or a particular reason that you had for doing something. In general, we use “especially” much more frequently than “specially”.
You can find a good explanation in the Oxford Dictionaries website.