Private businesses are usually owned by a family, or subgroup of a larger company. It can be either partnerships or sole proprietorships. While a public company is a firm that offers a part of itself for share to the public through Initial Public Offering (IPO) to raise investment assets, which means investors owned a part of the company’s profits and interests.
The key differences between private and public companies:
1. trading of shares: a private company can trade its private shares to a few private shareowner or investors while a public company stock is traded publicly on the stock exchange. The stocks of a public company are traded on stock exchanges.
2. number of investors: for starting a public company: at least 7 members are needed. For a private company: minimum 2 members.
3. reporting requirements: Public companies have more reporting requirements (ex. file quarterly financial reports, annual reports, etc.). Private companies have no reporting requirements.
4. access to capital: Public companies have immediate access to high amounts of capital, while private does not.
5. valuation considerations: Public companies are easier for business analysts and investors to assess than private companies.
6. risks.

It is hard to tell which type would success as it depends on the company. Many privately held companies earn much more than public companies.

Example of private companies:
– Chik-fil-A
– Mars Inc.
State Farm
Dell
Example of public companies:
– Google Inc.,
– Chevron Corporation
– Procter & Gamble Co.

Refrences: quora, investopedia, corporatefinanceinstitute, allbusiness