Paycheck - It is the amount of salary one is getting while being employed usually at those establishments which work on a sole proprietorship or partnership firms etc & do not get into the legal aspects of a private limited company. Even though quite a few pvt. compnay also pay salary by cheques. At these establishments payment of salary & wages are paid through cheques which can be fortnightly or monthly & they are not very structured & do not have some statutory deductions. Lot many times it is also used as a jargon to express the amout of salary what one can draw or is drawing in a employment.
PaySlip - It is a documentary proof that a person is employed in a company & drawing a certain amount of salary where usually salary slips contains all the breakup of the salary under different heads as well as other heads like "income tax deduction", "PF deduction" etc.