throws tells others that this method can throw an exception. Think of it as documentation. Checked exceptions must be part of a method's signature.

throw actually throws the exception. (throw new Exception(); first creates a new exception instance and then throws that instance. You could use two separate statements: Exception ex = new Exception(); throw ex;) You can throw both checked and unchecked exceptions with the throw keyword.

void checked() throws Exception { // required, Exception is a checked exception
  throw new Exception();
}

void unchecked() {
  throw new RuntimeException(); // throws not required, RuntimeException is not checked
}

void checkedCall() throws Exception {
  checked(); // throws required, because "checked" can throw a checked exception
}

void caught() {
  try {
    checked();
  } catch (final Exception ex) {
    // throws is no longer required, because the (checked) exception is handled within the method itself. It never leaves the method.
  }
}

Regarding your updated question: you don't throw new ArithmeticException, because it is thrown by the JVM when it tries to apply the / operator. You can throw it manually if you want to, but that's just redundant code.