Consecutive vs. Successive

Consecutive and Successive both apply to objects which follow one another without interruption or break.

Consecutive stresses the immediacy in following, regularity or fixedness of order and the close connection (as in time,space or logic)of the units ,consecutive days, weeks, years,...:

- He named the numbers from one to ten in consecutive order.
- She was absent from the office for sixth consecutive day.
- My brother won five consecutive races/games.

Successive
Happening or following in order immediately one after another without interruption regardless of differences in duration, extent or size or of the length of
the interval between the units:

- She weakened progressively by successive illness.
- It was his fourth illness after coming back to the country.
- He remained champion for the fifth successive year.

Here is an excerpt from the Merriam- Webster's Dictionary of synonyms :

One would speak of nine, ten. eleven as consecutive numbers since they follow one another in immediate and regular order ,but of flashing the successive numbers three,eleven and nine on a screen since the order would then be neither immediate nor regular;one would speak of successive (not consecutive) leap years since the order though regular is not immediate and of successive strokes of a piston since,though immediate, it need not be regular (the most important cause... has run throughout post-Conquest history like a consecutive thread - Coulton) (the product of the successive labors of innumerable men- Munford) Consecutive is also applicable to a person or to thought that manifests logical sequence (consecutive thinking absolutely requires personal initiative -Eliot)

References:
Merriam- Webster Dictionary of Synonyms and Antonyms
Longman Activator Dictionary