Overall, yes consecutively and successively are equivalent.
On closer examination, there is a slight difference though. In consecutively, there is no gap. In successively there is just some order.
For instance "in close/short succession" is sometimes preferred to "in succession" to emphasize consecutiveness.
1, 2 and 3 are consecutive numbers
1, 2 and 4 are successive numbers but they are not consecutive.
So for non discrete quantities you would probably prefer "successive" (as in "successive events") and for discrete quantities with no gaps you could use "consecutive" as in "2 consecutive days".