honed is more of a satin look but will still show surface imperfections. It is made buy sandblasting or brushing polished surfaces to various extents. It may also be called 'leathered' depending on the amount of distressing done to the surface. Because the distressing is an extra step, there may be an upcharge associated with this finish, and the upcharge will vary by stoneyard.

Your preference will dictate what you choose. Natural stone will have different appearances and may not even seem to be the same stone when the surface is treated in different ways, so make sure you get samples of the stone with the surface treatment you want before committing.

Some natural stone surfaces such as all stones called marble and quartzite (which is natural quartz, not synthetic surfaces which are mostly acrylic) will etch under normal kitchen use (especially with acids like tomato, citrus, vinegar, ketchup, mustard, salad dressing, etc.) so some people like to get them honed because the etching is less apparent-though still there, with this finish.

The basic properties of the stone you choose will not be changed by the finish.

Polished marble has a shinier finish. I've heard that polished stone may be more resistant to staining (vs. etching) than honed, as honed is more porous, but I can't attest to this, as my marble counters are honed.

This link explains that honed and polished marble both undergo a sanding process, but the polished stone is buffed to a shine.