If they are the same volume then there's probably not much in it. But I would imagine the pail to have more volume than the barrel. For primary fermentation, you need about 1/3 additional volume in the fermentor as headspace. So if you are fermenting 5 gallons, you should aim for a fermentor that's 7-8 gallons in size.
The headspace is necessary since the yeast can form a large head of foam (kraeusen) that will take up much of the headspace. (It can even overflow out the airlock if you've a particularly active ferment.)
For long term aging of high abv beers, wines and meads, the opposite is true. Then you want a fermentor that is as close as possible to your batch size to reduce headspace to the minimum. This reduces the amount of oxygen in contact with the beverage, which mitigates oxidative staling.