the subsequent thing = the thing that comes next or after another thing e.g. I met a student for a trial lesson. He decided to work with me. In the subsequent (=next) lesson, we reviewed one of his documents. Consecutive things are things that happen in a sequence, in a row, in a series, one after the other e.g. I met a student for a trial lesson. He decided to work with me. We reviewed five of his documents in five consecutive lessons. (= lesson 1, then lesson 2, then lesson 3...) In the second example, I could replace "consecutive" with "the subsequent" and the meaning would be very close. However, the idea of a series would be less emphasised. In the first example, we cannot replace "subsequent" with "consecutive".

“the subsequent lesson” would mean the very next lesson but “a subsequent lesson” would mean any lesson that comes after, not necessarily the next one. “Subsequent” just means “coming after”. “Consecutive” is used in math and that is a good way to illustrate its meaning.. For example, 1, 2, 3, 4, and 5 are consecutive integers. They are immediately in a row without any breaks or gaps.