In practical terms, USPS package shipping rates are typically 1/2 to 1/3 of those of an equivalent service from a private carrier. It's often cheaper to send something via USPS 2-day than via UPS/Fedex ground service. Their bulk rates are even cheaper - Media Mail is usually the only cost-effective way to ship textbooks for example. And Fedex/UPS simply have no service that competes with first-class or second-class mail.
In fact USPS's service is so effective that UPS and Fedex actually piggyback on their service to save money. Fedex and UPS will get the mailpiece to the local post office and the post office will handle the last-mile delivery. This is called SmartPost service and I can tell you from experience that the private carrier's involvement significantly worsens the quality of service. It often spends multiple weeks in the hands of the private carrier and I've noticed significantly more damage to mailpieces with it. They also provide subpar customer support on these packages, you cannot refuse delivery or do other basic postal operations with these packages. The opinion of the private carriers essentially seems to be that the blame will fall on the postal service so quality of service is irrelevant.
Nobody is perfect, and all carriers are subject to variation in the quality of the last-mile delivery. If your USPS or UPS or FedEx guy is particularly terrible then you will get damaged packages.