"Одинаковый" refers to two and more objects that look similar, alike. E.g. У всех девочек в классе были одинаковые платья. All the girls in the class had similar dresses. "Один и тот же" refers to one object (the same, one and the same). E.g. Она была такой бедной, что носила одно и то же платье в течение года. She was so poor that she was wearing one and the same dress during the year.

These expressions have the same root "ОДИН" which means ONE. They are, in fact, adjectives and change the form according to the gender of the name that follows. Ex.: один и тот же компьютер, одна и та же еда, одно и то же событие. That means we have only one object (though it might be something not material, like a thought, for ex.) Одинаковый/ая/ое/ые - this word supposes that we have 2+ objects to compare. Ex. Эти ботинки одинаковые (that means these shoes are both left, or both right)))

один и тот же (одна и та же, одни и те же, одно и то же) - the same одинаковый (-ая -ое -ые) - identical

Yes, previous answer is accurate. It depends on context if you translate from English. один и тот же - the same actor plays both roles. Spiderman and Peter Parker are the same person. одинаковый - all spoons from one set are the same. Like identical.