Arguably, it could be said that a business is the most direct/pure manifestation of economics. Arguably, again, it could be said that a government is the most direct/pure manifestation of politics.
A business, an entity, is a means through which a stakeholder's interests are represented. A government, an entity, is a means through which a state's interests are represented.
This question simply inquires about a specific level of abstraction among the following (from macroscopic to microscopic): private versus public, economics versus politics, businesses versus governments, and stakeholders versus states.
(Note: I am open to different orders of the above abstractions, or any additions. However, I am fairly certain that is the correct arrangement.)
(In regards to intent) Besides the level of abstraction, on a more sociological and psychological level, they are different degrees of self interest versus selflessness. A government, most types, would be more utilitarian and have the interests of more people in mind than just the immediate stakeholders of a business.
(In regards to consequence/impact) However, it is nonetheless noteworthy that it is possible for each to solve the same problems. This is because a hypothetical model of future governance (which is seemingly where the world is heading at the moment) is that governments would not really exist and the world would be entirely run by businesses (specifically, large companies/corporations).
Going off of the hypothetical ruling of the world through businesses, a key distinction could be said that a government has what is called enforcement. Specifically, physical prowness through military, police, etc. If anything, that ability to attack or defend in the physical sense is what distinguishes the two. However, would private security for business not count as a form of this..?