zvlášť is indeed used also for separately, independently (samostatně), but it has more meanings, for example: especially, strongly, special for this purpose
zvlátní is a bit different. First it is an adjective, but the meaning is somewhat different and again there are several meanings: strange, exceptional, extraordinary, special, weird and yes, even separate
For example zvláštní jednotky = speciální jednotky = special units (forces), zvláštní náhoda = a weird/extraordinary coincidence, zvláštní pokoj = separate (bed)room (but in a different context a strange/weird bedroom
The actual meaning depends on the context (as usual).
See, for example: https://en.wikipedia.org/wiki/Czech_declension#Adjective. And it is bound to be covered in the Tips & Notes, which are available in both desktop and mobile browsers.
Those are both "soft" adjectives, and the soft adjectives' declension differs from that of hard adjectives. See, for example, Soft Declension at this handy site: https://en.wikipedia.org/wiki/Czech_declension#Soft_declension.