Climb vs conventional cutting: what's the difference and which should I use?

The difference

It's about whether the flutes' rotation moves WITH or AGAINST the feed direction at the cut:

Conventional: the flute enters thin and exits thick, so chip thickness grows through the cut. The bit pushes against the material, and cutting forces load the machine steadily.
Climb: the flute enters thick and exits thin. The bit tends to pull itself along the cut, which on a machine with any backlash or flex can yank the cutter forward ("self-feeding"), and on handheld routers is dangerous. CNCs hold position, so climb is generally safe there, within rigidity limits.

The official documentation says the choice of Climb or Conventional is largely dictated by the material being machined and the tooling.

Climb usually leaves the cleaner edge (fibers are sheared downward into the cut, so less tearout and fuzz) but can be slightly worse for dimensional accuracy on flexy machines (deflection plus any backlash).
Conventional is more accurate and stable (forces oppose feed, with no self-feeding) but can lift fibers at the edge.

Climb for profiles is commonly used and is fine for most CNC work in wood. Check which direction your toolpath form is set to, because the default can depend on the strategy and your saved settings.
A common combo (best of both): rough with climb, leaving a small allowance (about 10% of bit diameter, shop advice), then a conventional finish pass at zero allowance, for a clean edge and accurate size. This is the same two-pass trick from the fuzzy-edges entry.
Flexy or belt-driven machines cutting hard material: prefer conventional for accuracy.
Pockets: direction matters less (edges get a profile-style finish pass around the boundary anyway), so leave the default.
Aluminum and plastics: conventional for stability on hobby routers, with a light climb finish pass for edge quality if the machine is rigid.

If one side of every cut is fuzzy and the other clean, you're seeing direction plus grain interaction. The two-pass combo fixes it.
Reversing the cut direction also reverses which side needs the lead-in, so re-check lead and ramp placement after changing it.