While every company is different, there tend to be significant contrasts between those that are privately owned and those that are publicly owned and traded on the stock market. Different management structures, different compensation packages and different routes for career advancement are among the dissimilarities. However, it's important to remember that there are exceptions to these general rules of thumb.
Public Vs. Private
While public companies are always corporations, private companies can be corporations, limited liability companies, partnerships or sole proprietorships. While private companies are usually smaller, that is not always the case. Google, for example, is a private company. Similarly, while there are many private companies that pay their employees poorly and have poor working conditions, in Forbes's 2017 list of the top 10 companies to work for, all but two are private companies.
Autonomy Vs. Structure
In private companies, there tends to be fewer policies and fewer levels of management. This can mean faster decisions and less micromanagement, with less red tape. However, less structure isn't always positive, particularly for employees who prefer stability and clear-cut policies for almost every situation. If you like the idea of picking up a project and seeing it through from start to finish, you're more likely to have this opportunity in a private company. If you want to know that you have a team around you and won't be asked to do more than you can handle, a public company may be a better option.
Public companies, which are usually larger and have more management positions than private firms, can usually offer faster promotions. They also tend to have more resources to help employees train and further their education while on the job. If you work for a private company, you may have to wait for your manager to retire before you can become a manager yourself. In a public company, not only are there more managers above you, there are more opportunities in other departments.
If the size of your paycheck is the key decision factor for where you want to work, you should probably aim for a private company. Most privately owned companies pay better than their publicly owned counterparts. One reason for this is that, with many exceptions, private companies aren't as well known, so they need to offer better incentives to attract the best employees. Private companies also tend to offer more incentive-based pay packages.
Private companies tend to have higher turnover rates than public companies. If you are working on a project, you are more likely to come in on Monday to find a team member has left than at a public company. If you are considering a position with a private company, ask about its turnover rate.