This is the most common question in hiking footwear, and the honest answer is that trail runners have won for the majority of hikers. Thru-hikers on the AT, PCT, and CDT overwhelmingly wear trail runners now, and weekend hikers are following suit. But boots still have a place.
Choose a trail runner if most of your hiking is on maintained trails, you carry a pack under 25 lbs, you value speed and agility, or you hike in hot conditions. Trail runners dry faster, weigh less, and cause fewer blisters. The Brooks Cascadia 19 and Altra Experience Wild 3 are excellent starting points.
Choose a hiking boot if you regularly carry loads over 30 lbs, hike on loose scree or talus, need ankle stability due to prior injuries, or hike in consistently wet or cold conditions. The ankle collar, stiffer midsole, and waterproof membrane of a boot like the Keen Durand 2 or Oboz Katabatic earn their weight on rough terrain.
The middle ground is a lightweight mid-cut boot like the Scarpa RUSH 2 PRO MID GTX. It offers ankle support and waterproofing at near trail-runner weight. For rocky terrain that doesn't quite warrant a full boot, approach shoes fill the gap with sticky rubber and precision fit. Not sure which side of that line you are on? Our approach shoes vs. hiking shoes comparison sorts it out by terrain.
Whatever you choose, fit matters more than category. A well-fitted trail runner will outperform an ill-fitting boot every time. Size up a half size, try shoes on in the afternoon, and always test with your actual hiking socks.