In case of wage employment, the employee gets to work for an external party for a particular wage rate as decided by the mutual consent between him and his employer. The employee working under wage employment has no authority to make decisions regarding the business. Also, he has no financial risk associated with the business.
On the other hand, a person in self employment works for his own business or as a freelancer for some other business ventures. He has all the rights and controls to make decisions and terms of his employment. Also, in most scenarios, self employed person has a financial risk of losing the investment incurred by him.