Dear student,
In short, the answer is no. Burning of wood is not cleaner as natural gas. The simple reasons can be obtained by comparing both. After burning wood, you find ash, which must be removed at regular intervals. At the same time, natural gas doesn't produce any Ash or particulate matter. There are fine paritculate matter(PM) that are produced as a result of burning wood. This contributes highly to pollution, as it causes asthma and breathing problems. The amount of heat produced by burning a unit mass(say 1kg) of fuel is called calorific value. Wood has much lower calorific value than natural gas. That is wood produces less heat than gas. Smoke produced by woods are the major reason for air pollution. As you must have seen, natural gas burns with much less smoke. Cutting down trees for firewood increases the problem. Natural gas is easy to transport and use. Wood will have certain amount of water in it, even if it is dry. Burning wood will release water vapour, which is a greenhouse gas. This leads to a phenomenon called global warming. In overall, natural gas is much better than wood.
Thank you.