Successive can be used to mean things that are following a certain order or series of steps; or things that are continuously occurring or occurring in succession.
1. It was his third successive win in the tournament = He has won three times without losing once in the tournament
2. This is the fifth successive month without rain = It hasn't rained for five months in a row
3. That's his sixth successive shot that hit the mark = He has successfully hit the target six times without failing
Succeeding, on the other hand, is used to mean something that comes after or come next in certain order.
For example, when you say "Adding sugar is the succeeding step after putting in the cream in this recipe," that means that in the series of steps for that recipe, you must first put in the cream before you add the sugar. Or, when you say "he succeeded his father's throne and became king", this means that he became the king after his father.
Lastly, subsequent is very similar to succeeding, but with subsequent, there is the element of time.
"Many collapsed from heatstroke due to the high temperatures. But, more damage was caused by the subsequent forest fires."
The statement above describes a series of events in time. First, high temperatures caused many people to collapse from heatstroke, then, after that, forest fires occurred.
If we're not being strict about it, succeeding and subsequent can be used interchangeably since they're considered synonyms.