They're both quite formal words but consecutive is more specific.
Consecutive means "one after the other". For example, "i ate two cakes consecutively". That means you ate one cake and then immediately began to eat the next one. The alternative would be to eat them simultaneously (which would be messy!) or just say "I ate two cakes" which could just mean you ate one and then had the other a few minutes, hours, or days later
Here is an example of a formal use of the word consecutive in legal language. https://www.nolo.com/legal-encyclopedia/concurrent-consecutive-sentences-double-punishment.html
Subsequent just means "at some later time" but there's less emphasis on the immediacy. For example, I have an Israeli stamp in my passport which looks good but might cause problems in any subsequent visits to certain other countries in the middle east".
Now that I've written this, I think there's another difference between the words. In the "consecutive", two or more events can be consecutive (the team played four consecutive games of football) but in the case of "subsequent", you're definitely talking about something that happened later ("the star player was injured during the second game so he didn't play as well during the subsequent games")