Monster drops
Monster drops are items that you sometimes get out of the defeated monsters. For example, Poptop drops Sharpened Claw.
Hunting
Defeating the monster with a "hunting weapon" (all bows, all spears, all shortspears, most crossbows/styngers) will yield "Hunting" drops instead of normal drops.
Hunting drops have a higher chance of meat (Raw Steak, Raw Ribs, Raw Poultry) and other animal products like Leather or Venom Sample. However, they have a lower chance (if any at all) to contain weapons, musical instruments, etc.
Note that a hunting weapon must land a final strike. If you hit the monster with Irradium Spear and it survives, but then dies to the ticks of Radiation Burn effect, then it won't count as a hunting kill. But if the same Irradium Spear kills the monster with a strike, then it counts as a hunting kill.