Splitting captured monsters
Splitting captured monsters will result in components which can be used to upgrade existing weapons and shields and also craft new weapons, shields and accessories using recipes.
Keeping captured monsters
Not splitting monster parts means that you can sell the captured monster to different shops which will result in more money than splitting them would. Selling captured monsters will also unlock new items for purchase over time.
What should I do then?
With no components you can't upgrade your equipment, so its generally a good idea to split captured monsters until you have at least around 10 of the associated components before you start to sell them instead.
Also, some components can be rare which is also a reason to initially prefer splitting.
As stated above, besides money there is another reason to sell captured monsters. By selling captured monsters, you will cause new items to appear in shops over time. Each monster is associated with different categories and when sold, points are added to these categories.
By accumulating enough points in a category, new items are unlocked, called trade goods.
For a list of unlockable trade goods, please see this wiki page.
To summarize, splitting should be preferred initially for crafting and upgrading reasons, but selling the captured monsters will net you more money and the availability of new items in shops.