Which game does monster hunting well?

[Simlasa]

X-Files and Supernatural are monster hunting shows.

I disagree that the 'tonal difference' is always or necessarily present. Those shows have elements of both.
X-Files most certainly had episodes that were full on horror... with either or both of the main protagonists in danger. They were the hunted.
One of it's main influences was Kolchack The Night Stalker and that was both... with Kolchak being a barely capable monster hunter who often ran away in fear before managing to (sometimes inadvertently) win out over the creature.
Supernatural... certainly a 'monster hunter' show... but also horror in the early seasons. Later the bros became seemingly invulnerable.

Call of Cthulhu is a 'monster hunter' game as well... but also solidly horror.

 

[Retros_x]

Call of Cthulhu is a 'monster hunter' game as well... but also solidly horror.

But come on, would you really suggest CoC as a fitting game for the vibe OP looks for? Cosmic beings driving PCs insane sounds really not like the vibe of Supernatural meets X-Files.

I can also wholeheartedly suggest Monster of the Week. Its EXACTLY the vibe, as overgeeked commented. I played in a mini-campaign (with the setting of 80s cold war GDR swampy hinterland of Brandenburg/Mecklenburg) with a quite similar targeted vibe and it was lots of fun.

 

[GrimCo]

I would go with Hunter:The Vigil. Or just plain old WoD core rulebook (2004 White Wolf edition). We played long, Supernatural inspired campaign using it and it was blast.

If you want hunters to be more normal people stuck hunting monsters, core WoD and Tier 3 Hunters are way to go ( regular joe humans, with no broader organisational support or supernatural edge). Compacts and Conspiracies ( tier 2 and 3) are more organised, better funded, some are tied to Church (Maleus Maleficarum), some to goverment ( Task Force Valkyre), some are straight up supernatural in it's own right (Lucifuge).

It's older game, but it get's job done.

 

[Autumnal]

Savage Worlds would do it well with just the core book, I think.

 

[overgeeked]

Savage Worlds would do it well with just the core book, I think.

Almost certainly. But if people want the extras there’s also the Horror Companion and the Rippers setting.