"My aunt is an actress."

There is no real difference in meaning. «Актри́са» is borrowed from French actrice (via Russian актри́са). «Акто́рка» is formed from «акто́р» using the Ukrainian suffix -к- often used to create feminine professions (compare other words like that: журналі́ст '(male) journalist' — журналі́стка '(female) journalist', юри́ст '(male) lawyer' — юри́стка '(female) lawyer').

Having two words, one borrowed and one created using native means, is a pretty common situation. Compare English pairs like automobile/car, autumn/fall, where the former is borrowed from French and the latter is a native word.