The people who say you pay less on taxes have no idea what they're talking about, or how taxes work. Ignore them please. Get paid under whichever option is better for you, it doesn't matter for taxes.
You may, depending on how the school does payroll, have slightly more or less withheld depending on how you're paid. This is pretty common with bonuses actually, where the payroll system will withhold as if the bonus is the regular salary. At the end of the year however, when you file your taxes it's (extremely oversimplified) taxable incomes times tax rate (really, really, really oversimplified. There's not one tax rate that you pay, it's actually different based on how much income you have and I'm slightly feverish from a vaccine so don't want to type how the graduated income tax works).
Anyway, total tax minus what you withheld is what you either have to pay or get returned to you, depending on if you over or under withheld. If you have way more than your tax liability withheld then you're giving the government an interest free loan. If you withhold way less then you may owe penalties. But it all comes down to the totals and get trued up when you file your taxes.