Paycheck - It is the amount of salary one is getting while being employed, usually at those establishments which operate as sole proprietorships or partnership firms, etc., and do not involve the legal aspects of a private limited company. Even though quite a few private companies also pay salaries by cheques. At these establishments, payments of salaries and wages are made through cheques, which can be fortnightly or monthly. They are not very structured and do not have some statutory deductions. Many times, it is also used as a jargon to express the amount of salary one can draw or is drawing in employment.
PaySlip - It is a documentary proof that a person is employed in a company and drawing a certain amount of salary. Usually, salary slips contain all the breakdown of the salary under different heads as well as other deductions like "income tax deduction," "PF deduction," etc.