to a distinctly greater extent or degree than is common
"he was particularly fussy about spelling"
"a particularly gruesome attack"
"under peculiarly tragic circumstances"
"an especially (or specially) cautious approach to the danger"
For a special purpose, person, or occasion.
"The cake was specially baked just for her birthday party."
In plain English: Specially means doing something for a specific purpose or person rather than just generally.
"We specially ordered this cake for your birthday party."
Usage: Use specially when an action is performed specifically to meet the needs of a particular situation or individual, such as building a house specially for your family. Do not use it interchangeably with especially, which emphasizes one item over others in a group; instead, reserve especially for highlighting exceptions or degrees within a category.
The word specially comes from Middle English, where it was formed by adding the suffix -ly to the adjective special. It entered modern usage as an adverb meaning "in a special manner," directly extending the original sense of its root word.