Learners always have doubts when using specially or especially. Both words are very close in pronunciation and meaning, but have quite different uses.

According to Inglês na Ponta da Língua and the Oxford Dictionary:
1) Especially means "particularly", "mainly", "exceptionally" and isn't used to start a sentence.
- She loves all sports, especially swimming.
- I especially like sweet things.
- We chose this painting especially for you.
- I would especially like to bring up one matter…
- Chocolate, especially white chocolate, was João’s weakness.
- We were especially happy to see you.
2) Specially is closest in meaning to "for a particular purpose" and is often followed by a verb in the part participle, such as "designed", "developed", or "made".
- This course was specially designed to meet your needs.
- She has her clothes specially made in Paris.
- This glass has been specially made for use in cars.
- The cake had been specially prepared for the occasion.
It's important to highlight, however, that Oxford brings a note saying that in British English both specially and especially are used the same way, being specially less formal.
- I bought this especially/specially for you.
- It is especially/specially important to remember this.
Although some speakers say there's no big deal while using one or the other, proficiency exams and more grammatical observers may surely consider this slight difference in use of specially and especially. So, why not using it properly since ever?
Source: Oxford & Inglês na Ponta da Língua