Due to supply and demand, it is cheaper to move during the week than during a weekend. More people will always try to schedule their move on the weekend, which means moving companies are likely to charge higher rates on those days. Weekend moves will almost always cost you more money because the moves have full schedules. Sure, the kids are at home, but you'll probably get a babysitter anyway, so don't get in the way.
Also, there may be no traffic at peak times, but most of your neighborhood or apartment building is likely to be off work. This means you can have a lot of people looking through their blinds and checking your furniture. There's nothing necessarily wrong with moving on a weekend, but you should know that it's not necessarily the most attractive option, if you stop to think about it. The fees charged by a moving company are based on their practices.
In most situations, it's cheaper to get around during the week than during the weekend. There is sound reasoning behind this, which, once you think about it, will make more sense and may make you want to take advantage of the discount. The weekend is often a popular time to schedule a move because that eliminates the need to take a day or two off work. That means that weekdays often have more vacancies available, whether you hire professionals or do the work on your own, so that can amount to cheaper prices.
While the most popular day to move is Friday, this doesn't mean it's the best day to move. If you can take a day off work, you can move on a Friday and unpack your bags over the weekend, or move out on a Saturday and unpack your bags on Sunday and Monday.