Many employers (but not all) are diligent in making certain that employees who are not working on the holiday still get paid time off for the day, and that is how it should be. For those who work on the holiday, relatively conscientious employers (but, again, not all) provide their employees with time and a half "holiday pay." While it is better than nothing, it is a significant disadvantage to those who work the holiday as opposed to those who do not.
Let's take this week for an example. On a "normal" week, a full-time employee would earn 40 hours of pay for 40 hours of work. For the sake of convenience, let's say that this was for five eight-hour days--Monday through Friday. Now, the Fourth was on Monday, so let's first look at the employee who gets the day off, but no additionally scheduled days; this employee gets 40 hours of pay for 32 hours of work. In comparison, the employee scheduled to work on the holiday gets 44 hours of pay for 40 hours of work. Essentially, the second employee is working the holiday for half pay.
To drive the point home a bit more, let's take that first employee (who gets the holiday off) and say that they (yes, I'm using a gender-neutral plural for a singular pronoun--I said this day was coming almost 20 years ago) have a job that still needs them to put in five days of work, so they decided to come in on Saturday. As it is part of the work week, they should (if the employer is ethical) get time and a half for that Saturday, but let's first say that their employer is a jerk, and they don't; they would get 48 hours of pay for 40 hours of work. Even though the first and second employee are putting in the same amount of time, this first employee is being paid more. To clarify, with an ethical employer, that first employee should be making 52 hours of pay (due to the overtime) for 40 hours of work because the holiday counts as hours of work.
In short, you are better off not working on a holiday.