I think the reasons why people want to hunt using spears are the similar to the reasons for archery and crossbow hunting: it's about the challenge of hunting with weapons that are - at their core - a lot more primitive than what would be available to hunters.
This could be out of appreciation of indigenous hunting techniques, or because the hunters value the additional challenge, the practice and experience needed - but this is pure speculation at this point and the reasons will vary from person to person.
It's arguable whether these are rational reasons - for some of them are certainly emotional ones... But if you look into the past there certainly have been rational reasons why hunters hunted using spears instead of bows:
- To some societies bow and arrow were simply not known.
- Spears are a lot easier to manufacture than bows.
- In some societies spears were much stronger than the available bows in terms of force/penetration and were thus able to perform kills of larger animals quicker. (Not wanting to start a discussion on the potential strength of bows here - I'm aware that there were/are some bows that are strong enough to kill pretty much any animal out there...)
- If you want to attach a line of any kind to your projectile (say hunting/fishing in water) the spears allow for this a lot easier than bows, where the line tends to become tangled in the bow/string.
- A spear can be used in self defence, where a bow is rather useless in such a situation.
On a side note:
Recently spear hunting has lead to a bit of controversy, see for example this story where a filmed spear hunt of a baited black bear led to spear hunting being banned in Alberta.