How do I know the difference between a sore lower back and an strained lower back?
If you have to ask, you should probably ask a doctor.
But, as a quick answer, low back muscle soreness from a workout feels like any other muscle soreness. If you have any shooting/stabbing/burning type pain, if you have pain that radiates down your legs, or if your back is unreasonably stiff and it is painful to walk, sit, or stand, then it is possible that you have a more serious injury. I am not a professional, but I do have a herniated disc to derive experience from.