It is extremely useful to collect the dragon’s breath during the fight because it can be used to make lingering potions, which can help if you want to fight the dragon again, or if you want to fight the wither (if you chose to fight the dragon first).
To collect the dragon’s breath, wait for the dragon to perch and do its breath attack or shoot a dragon fireball, either of which puff out into lingering purple clouds. The only difference is that the dragon’s breath comes instantly from the breath attack, whereas you must wait for the fireball to slam into the ground. To collect it, simply hold out an empty bottle and scoop up the clouds in the same way you would fill a bottle with water. However, you should avoid making contact with the clouds, because the clouds cause damage. Armor cannot protect you from damage from the dragon’s breath, because it counts as magic damage. The dragon’s breath is stackable, so you should collect as much as possible. Additionally, the leftover bottles from your used potions can increase your collection.