I'll caveat this by stating I'm not an expert in the area, but the obvious difference seems to be that I can keep growing more trees to burn, while the only way to keep burning coal is to dig up more of it. Burning a tree may release the same amount of CO2 as burning the coal, but the carbon in the tree was taken out of the air during the growing process, so the total amount of CO2 hasn't changed over the lifetime of the tree. Burning the coal releases previously stored carbon without a way to "re-store" it since new coal isn't being created at nearly the rate we're burning it.

Really burning the tree would have the same problem if you were burning them far faster than new trees were growing, but the idea is that you don't do that. The problem with fossil fuels is have no way of restoring it remotely as fast as we're burning it, so the net amount of carbon in the air has to go up.