The biggest difference between formal and informal leaders is that formal leaders are put into an official position, and informal leaders are generally looked up to, although they don't have an official position. Some people refer to informal leadership as having a command presence.
Formal leadership styles are typically seen in the management of most businesses and clubs, as they usually run off of a type of hierarchy in the group. This prevents confusion, makes it easy to determine who is in charge and makes delegating responsibilities simpler. A formal leader would be someone hired for the position, such as a manager at a store, or someone elected to the position title, like a president. They did not get those leadership positions through influencing others and having experience; the entire reason those positions were given to them were because the elective officials decided on that one person. This can be a bad thing, since there is no guarantee that the electing officials, or hiring managers, or that the people being elected or hired have any relevant experience or leadership skills.
Informal leadership style is a type of leadership that is gained through respect and trust. People who are informal leaders have something like a command presence, an innate knowledge about the tasks at hand or some other form of seniority. However, they don't actually have to have a title that denotes them as an official leader or manager. Someone that employees look up to could be the informal manager because he takes care of them and knows where the products go and the prices. This could cause problems because depending on the informal leader's personality, this can cause undermining the titled manager and cause distress and disrespect.