Which object hits the ground first, a bullet fired horizontally or a bullet dropped from the same height?

They will both hit the ground at the same time as they both start with 0 vertical velocity and are acted upon by the same gravitational force.

This however is in a vacuum and I'm not sure about the wind-resistance of bullets, but I'd say the difference would be negligible.

The dropped bullet will probably hit the ground first but I'm not entirely sure. Under gravity in a vacuum they would hit the ground at the same time as Hijakr (and Galileo before him) points out, but in air the long trajectory will cause it to spin through more air at a continuously changng angle. I say I'm not sure because air is a fluid and the fluid dynamics of the air around a bullet is quite complex.

According to this site, they will hit the ground at the same time

http://hyperphysics.phy-astr.gsu.edu/hbase/grav.html

If the gun fired ultra high velocity rounds then the one that is fired potentially has "further to fall" due to the curvature of the Earth.

Secondly, what you have in your hand (before being fired - ie a "round") is vastly different to the chunk of metal that leaves the barrel at many MPH.
A dropped "bullet" has a case that contains the explosive. When fired this is left behind in the gun and is not part of the projectile !