Handling:
Bolt action - simple
Double rifle - complicated. (picky on ammo, regulation of barrels is always a subject, reloading to be trained)
Range and effectivness:
Bolt action: short range + long range
Double action: short range
(For usual hunting distances, bolt rifle is usable up to 200 or 300 meters, as well as on 30 meters, while double rifle is 10 mtr - to optimistic 100 meters weapon)
Reloading:
Bolt action, work the bolt
Double rifle, break open the action, reload somehow.
Accuracy:
Bolt action - 1 to 1.5 moa, acceptable and expected
Double rifle - 4 - 5 moa, excellent (for double rifle standard). Anything better is lottery.
Reliability:
Double rifle with two triggers is like two rifles in one stock
Bolt rifle, is only one rifle. And if it jams, than it is jammed.
Price
Double rifle - good one cost more than any factory rifle
Bolt rifle, affrodable for middle class. High end factory bolt rifle, will be cheaper then any good double rifle.
Ammunition:
For standard calibers ammunition is much more available for bolt rifle
Shooting:
Bolt action, one after another. But two consecutive shots on bolt rifle are slower than on double rifle
Double rifle, gives two fast shots.
But 4 shots from both of them, comes within same time for skilled user, trained for fast reload.
Two fast shots as a main quality of double, are not as easy as it sounds just a trigger pull apart, because just like on bolt action, a shooter needs to handle the recoil, come back from recoil to align the sights and shoot again, and most likely between two shots the target will be moving.
It is not just a fraction of seconds between the two shoots, because things happen in between, just like with a bolt action.
Bolt rifle is rifleman's rifle. Put the scope, zero, shoot, hunt, and repeat
Double rifle, needs philosophy, skill, attitude, and aptitude, load development and dedication. Double rifle is a lifestyle of a hunter. It is very hard to achieve that. It takes time to get skill and all planets and gradients alligned.
Having said all that:
My experience with double rifle is limited. I tried double rifle from a friend, to test on a range It was in 30-06, extensively. I was not impressed. It was clumsy for reloading for me, and accuracy was disaster compared to any bolt action I know
Then I tried double rifle in 416 in hunting camp, on a range in Zim. With the idea to use in hunt. I finally decided for bolt action rifle.
With those two testings, I came to above conclusions, and ever since I keep away from double rifles.
In my two DG hunts (buffalo and ele), my success in relationship to rifle performance was simplicity and accuracy of bolt action rifle.
If I used double rifle in two of my DG hunts, my entire experience would be different, and how it would end up, I cannot say - because entire stalk and approach would need to be different.