Why are there two plural forms of trauma? How do traumas and traumata differ in origin and nature? Is one incorrect?
There are at least hundreds of words with multiple plural forms: staffs and staves, dice and dies, châteaus and châteaux, pike and pikes, cows and kine, millenniums and millenia, phalanxes and phalanges, mongooses and mongeese, and so on and so on. The reasons why multiple forms exist, and the cases in which one form is preferred to another, are widely varied.
In the case of trauma, the rare form traumata reflects Greek pluralization; trauma is a third-declension noun with an -at stem. There are many such words that were brought into English such as schema, stoma, stigma. But the regular traumas also exists.
In very broad terms, the more common the usage, or the more recently it has been popularized, the more likely the Anglicized plural is preferred; the more academic or specialized the usage, the more likely the traditional pluralization will be preferred. Classical pluralizations are also more common in British English than in American English.
Thus, we have philosophical schemata but database schemas, radio antennas but Formica antennae, presidential memorandums and royal memoranda. If you have more than one cherub in the sense of a chubby, rosy-cheeked child, you have cherubs. If you have more than one cherub in the sense of a certain type of four-faced, four-winged angel, you have cherubim.
I doubt you will find much use of traumata nowadays outside of medical or scientific literature, likewise blastomata, carcinomata, hematomata, sarcomata, and many dozens of other words. Before the 1960s, trauma was mostly found in medical or psychological literature; since then, its increase in uses and in the innovation of uses has favored the English, rather than the Greek, plural form.
Traumata (pronounced "TRAW-muh-tuh") is from the Greek plural; traumas was made in English using the English plural suffix -(e)s. Neither is incorrect. Traumata is not commonly used, or even commonly known, although you can find it in some academic works. A Google Ngram Viewer chart comparing them suggests traumas is more commonly used. But neither of the plural forms is particularly frequent compared to the singular form trauma.
There are a number of other words taken from Greek that can be pluralized by changing the -ma in the singular form to -mata, such as stigma, schema, lemma, comma, miasma, teratoma. These words were all neuter in Greek. (Reference: "Latin and Greek endings / plurals v1.1," uploaded to academia.edu by Catherine Ebenezer)