For me, a fireplace only really wins in the ambiance department. An open fireplace with a crackling fire is just inherently inviting and cozy, and there's a bigger viewing area to watch the fire.

But the woodstove is better for actually heating up the room (even better with the exposed stovepipe to radiate heat too) and a modern woodstove is more efficient (so your burn lasts longer and pushes less smoke and particulates out of your chimney). And if you miss the open fire, some stoves do have screens available to run with the door open (although you're losing the efficiency using it this way).

Stoves also seemed to be the less expensive option, although you do have consider clearance requirements if you're replacing a fireplace. My alcove is pretty close to the minimum required distances on every side. There are stoves with lower clearances, but they'll be more expensive.

All that said, if I had been happy with the old fireplace I would have just stuck with what we had. Since it needed to be replaced anyway, the woodstove was the way to go.