The room was specially designed to be used as a library. I was specially pleased with your gift. The speech was written specially for the occasion. I don't want to be treated specially. The cabin crew is specially trained and equipped to put the fire out.

We use specially to talk about the specific purpose of something: This kitchen was specially designed to make it easy for a disabled person to use. He has his shirts made specially for him by a tailor in London. Especially can also be used to mean ‘for a particular purpose’:

Specially most commonly means in a special manner or for a special purpose, as in or This was specially made or You were specially chosen for this project. Especially means particularly or exceptionally, as in That was especially helpful or This road can be dangerous, especially when it’s icy.