setError should not be a const, when you try to change its value it throws an error and reject the Promise.

let setError = "";

Because you are assigning to const setError, it causes runtime error and stop the function execution.

const cannot be re-assigned after initialization. Making it let solves the problem

let setError = "";

function validate() {
  return new Promise((resolve, reject) => {
    const exp = RegExp("w+@example.com").test("sfsdf@gmail.com");
    console.log(exp);
    if (!exp) {
      setError = "Email Not Valid!";
    }
    console.log("The value is" + setError);
    if (setError != "") {
      reject();
    } else {
      resolve();
    }
  });
}

validate()
  .then(() => {
    console.log("Resolved");
  })
  .catch(() => {
    console.log("Rejected");
  });

The second console.log is not firing because setError is a constant and cannot be changed. Changing const to let should generate the output you expect:

false
The value isEmail Not Valid!
Rejected