Is there any benefit of provoking a war, rather than just starting it?
You receive a lower warmongering penalty
Warmongering is an in-game term used to denote general aggression towards other civilizations, such as excessive waging of wars and capture of cities. It can have dramatic effects on diplomatic relations with AI-controlled leaders.
Simply speaking - if you are fighting a war (and especially when you are starting one) other countries start to hate you.
If you are in a defensive war, you won't get this penalty... that is until you will start conquering enemy cities, when this penalty will start kicking in. Also, this penalty depends on the era that you are in - in ancient times there is no penalty, after that it gets bigger with each new era.
There is this excellent guide (yes, it is for Civ VI but there are few differences between V and VI in this case)
The main sources of Warmongering are:
- Declaring a War
- Conquering a City
- Wiping Out a Civilization [...]
Or to be specific for the Civilization V: your warmongering score is calculated as:
City Capture: (1000 * est_num_cities) / (actual_num_cities * victim_num_cities)
Declaration of War: 250
est_num_cities is a static value that is dependent on the size of the map. It is set to 13 for Duel maps, 26 for Tiny maps, 39 for Small maps, 52 for Standard maps, 80 for Large maps, and 132 for Huge maps.
actual_num_cities is the total number of cities (including city-states) that exist anywhere on the map.
victim_num_cities is the number of cities a civilization controls before one of them is captured.
Warmongering score decays at rate of 5 points per turn. More info (with math) can be found here
As to consequences:
When you've racked up massive Warmonger Penalties and the World sees you as a threat, it is likely every Civ -- or at least those that still have power to do so, will Declare War on you at once. Little Civs you've angered in the past may even pitch in. It can be difficult to avoid, and you want to be the top Military power in those cases or your war effort will stall, and you'll find yourself on the defensive. Trade Routes with City-States and internal City Connections are your only options then, and your Cities will need access to unique Luxuries to make up for themselves. A City by default will cost you 3 Unhappiness + 1 for each Citizen in the City. source
It is smart to let them attack you.