I have a woodburning stove, for the sole reason that the people I bought the house from had it fitted. I light it only on especially cold nights; the rest of the time the gas central heating does a good enough job.
Here in the UK, I'm not aware of any significant impact on house insurance prices.
Reading around the web, I've seen anecdotal evidence that wood is cheaper than the equivalent gas -- but I'm not entirely convinced, and of course that would break if everyone started using it. With very occasional use, I've made the broken palette our washing machine was delivered on last most of the winter, so that was free
It does burn clean -- the smoke from our chimney is not visible. The manual says some stuff about recirculating the combustion gases so they ignite a second time before going up the chimney -- while simultaneously maintaining a flow in front of the window so that it doesn't soot up. The volume of ash is tiny compared to the open fires of my youth, and it can safely be chucked in the compost bin. And as has already been said, if it's from managed forests, its carbon will be re-fixed by the next generation of tree planting.