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".