What is the difference between mestizos and mulattoes?

Mestizo is a racial definition, while Mexican is a nationality.

Mestizos are people of both Native American AND white Caucasian ancestry; they are the product of interracial marriages between Spanish conquistadors and Native American peoples during the Age of Exploration.

Now, Mexican would be the term to define anybody born in Mexico, or from Mexican parents.

Therefore, you can have mestizos of Mexican, Guatemalan, Colombian or Peruvian nationality, and Mexicans of white, Amerindian, mestizo and even Asian ancestry.