They grant you valuable Research Points, which are needed for a huge number of activities in Astera, from unlocking new camps to purchasing rare items (look for the Argosy Captain later on in Monster Hunter World to see what I mean.) Investigate tracks! Even if you've already tracked the monster you're hunting, you should still make an effort to pick up tracks as you go on your way.You'll need the ore from these to forge weapons and armor. Almost more importantly, stop by Mining Nodes.These are important for crafting the items you need on quests.
This means: Pick up mushrooms, plants, bugs, and honey - especially honey - on your way. Though hunting a monster is usually the main goal of any quest, there are quite a few things to look out for while out and about. Read How to Capture a Monster for more information on how to do the deed. If that goal hasn't been met and it's close to death, it's better to keep working at breaking the horns before attempting to capture. For example, Diablos' horns need to be broken to acquire the horn materials. Some monsters require certain parts to be broken for certain rewards to drop. Each material has the same chance to drop, but instead of getting three carves from the monster's body, you get four or more 'capture rewards' on the reward screen with thes same drop chances. Capturing a monster is always better than killing it.