What is the difference between market failure and government failure?

A government failure and a market failure can be roughly compared to a person drowning and to a lifeguard who drowns trying to save that person.
Market failures can occur for many reasons, including inefficient delivery of goods, lack of supply to meet demand, drastic overpricing or underpricing, or overinvestment in aspects of business that fail to produce results. Market failures are typically the fault of businesses because of improper long-term planning or poor customer research.
By contrast, a government failure is an attempt to solve a market failure that obviously does not succeed. For example, the US government bailed out the automobile industry during the Great Recession: while this was not a failure, it could have been if the government intervention was not strong enough, if it was directed toward the improper channel, or if the government put in subsidies to keep businesses alive at the cost of efficiency.
Government failures are less common in market economies than they are in planned economies, where the state decides the output of industries. Even so, they are not uncommon in market economies like that of the United States. It has been argued that subsidies to boost the coal mining industry are government failures, since the cost of coal is higher than some other energy sources like oil and gas: the government is thus promoting inefficiency by funneling resources to coal.

Government failures cannot occur unless market failures have already happened. When the government involves itself in the economy in order to try and rectify an existing market failure, this can sometimes resolve the failure, or it can generate further inefficiency, such that the market failure is not resolved. In this situation, a government failure is said to have occurred. This may be because the government has altered prices, which has led to a lack of competition or to shortages or monopolies, or because the government has allocated too much budget to a failing company in subsidies, meaning that other companies are unable to compete, leading to antitrust issues.
A market failure is what happens when the balance between supply and demand is disrupted. This can be because one firm has developed a monopoly over an entire market, because resources have been incorrectly allocated and there are shortages, or because poor decision-making has resulted in excessive costs being passed on to the consumer (unfair price rises relating to manufacturing decisions or shortages).
https://www.economicsonline.co.uk/Market_failures/Government_failure.html