A bike backpack does make a difference, especially if you're cycling every day. The thing is, purpose-built cycling backpacks have a design that distributes weight better across your back, wider straps that don't slip, and usually a waist belt to stabilize everything while you're pedaling - something a school backpack just wasn't made to do. It's worth investing in because you avoid back pain and that discomfort you're feeling right now. If you want something more budget-friendly, any backpack with a waist belt helps a lot, but actual bike backpacks have some details like better ventilation on the back and pockets designed for your phone, keys, and stuff you use on your commute.