If something has been done specially for a particular person or purpose, it has been done only for that person or purpose. a soap specially designed for those with sensitive skins The school is specially for children whose schooling has been disrupted by illness
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"
Specially is used to mean more than usually or more than other things. What was specially enjoyable about that job? = particularly