If you climb with spikes and a lineman's belt, you can severely pull your groin, hurt your back, or a number of other hip/knee injuries should you fall with a spike dug in to the tree and the other comes out suddenly.
If you use an ascender, you're basically just out of luck and you're going to slap the tree and then fall.
If you've screwed in steps, count on getting some deep tissue bruises or bone bruises from hitting those.
I worked for a number of years as a tree climber when I restored log cabins and ran a tree service.
Climbing with a harness is dangerous for a number of reasons, not the least of which was highlighted above: dead or dying trees. If you don't know anything about trees, you can climb one that is dead in the middle or halfway up without realizing it.
Should a tree's canopy fall while you're strapped to the tree and split taking part of the trunk with it, you'll be crushed against the trunk by the weight of the falling canopy. A number of arborists and loggers have been killed this way from sawing trees the wrong way. It's a remote possibility, but you should know that it can happen.
Another way people get hurt is by not using climbing rope for their belay line. When regular rope gets a dynamic load (something falling that stresses the rope quickly) it pulls taut very quickly. This energy is transferred to you, your guts, connective tissues, bones, and your joints by the place where the harness is connected (hips). It hurts.
That's why climbing rope is made with a stretchy core. The core absorbs some of the shock by elongating.
It's not just the steps and bark that will hurt you. It's everything.
Tree climbing is the new black in hunting, but that doesn't make it less dangerous for neophytes.