Chewy cookies are made with more brown sugar than white sugar, which creates a moister cookie. More moisture means that the proteins in the flour have more time to absorb liquid and swell, resulting in a chewier cookie. Chewy cookies also have less leavening than crisp cookies. This combination of factors leads to a denser, moister cookie that spreads less while baking and holds onto its shape better.
Crisp cookies have a higher proportion of white sugar to brown sugar. This creates a drier dough, which allows the cookies to spread more while baking and results in a thinner, crispier cookie. Crisp cookies also have more leavening than chewy cookies, which helps them spread even further while baking.