The two words stem from the same agriculture-related root in Latin:
culture (n): mid-15c., "the tilling of land, act of preparing the earth for crops," from Latin cultura "a cultivating, agriculture," figuratively "care, culture, an honoring," from past participle stem of colere "to tend, guard; to till, cultivate" (see colony) ... The figurative sense of "cultivation through education, systematic improvement and refinement of the mind" is attested by c. 1500. Century Dictionary writes that it was, "Not common before the nineteenth century, except with strong consciousness of the metaphor involved, though used in Latin by Cicero."
cultivate (v): by 1650s, of land, "till, prepare for crops;" by 1690s of crops, "raise or produce by tillage;" from Medieval Latin cultivatus, past participle of cultivare "to cultivate," from Late Latin cultivus "tilled," from Latin cultus "care, labor; cultivation," from past participle of colere "to cultivate, to till; to inhabit; to frequent, practice, respect; tend, guard," ... Figurative sense of "improve by labor or study, devote one's attention to" is from 1680s.
For this specific definition (related to arts and literature), and use as a past participle adjective, "cultivate" is older, but nowadays I suspect "cultured" is more common.
"Cultivate" seems more common as a verb, possibly because it's impossible to separate this figurative meaning from the more common agricultural use. When someone is "cultivated", it's as if they are a carefully tended field, either by their own action:
She has cultivated an interest in medieval poetry.
or edification by others:
In past centuries, schools routinely cultivated a knowledge of Latin and Greek in their young charges.
In contrast, "culture" is not often used for general agriculture. Instead the verb is common for things like bacteria, or which rely on bacteriological action, such as yogurt and other fermented foods. So "cultured" seems more related to the noun "culture" referring to arts, literature, etc. Rather than the result of a process, "cultured" seems to imply a state of being. For example, this sentence:
I could have had no more charming or cultured travelling companion.
implies the person already has culture, rather than has been cultured. However, this is not always the case, for example this sentence:
When he spoke, his cultured voice was suave as ever.
can imply that he has intentionally made his voice sound "cultured".
To sum up: Both can be used to imply "knowledgeable of the arts, etc." but it's important to consider the source of the two words, and understand the nuances.
Side note: Because both of these refer to growing things, it's possible to use this for (mean-spirited) humorous effect:
Alexa: I would say he's a profoundly cultured man.
Benjamin: You mean he knows a lot about the arts?
Alexa: No, I mean he smells like old yeast.
Alexa: He always has a cultivated expression on his face.
Benjamin: You mean, like he has some deep knowledge he wants to impart?
Alexa: No, he's got these deep furrows across his forehead, like it's been plowed.