The advantages of non-motorized treadmills are they are lot more cheaper than the motorized ones. They come in smaller sizes, so it is easy to place them in smaller places. The non-motorized treadmills weigh a lot less than the motorized ones and easy to fold and place them in a corner.
For you, if the cost is the main reason going for a non-motorized manual treadmill is recommended. But for all other reasons it is better to go for a motorized treadmill. If you are basically althletic and your body is in good shape and you need a treadmill for some time management reasons, you can go for a non motorized manual treadmill. But if you are not in good shape and not used to running, then it would be easy for you if you go for a motorized treadmill.
Non-motorized treadmills have belts which move when you push them backwards. Practially you will be running by pushing the belt backward and this may give strength to your legs but they may also get injured if your legs are not already in good shape. Treadmills are designed for running and improving your stamina not strengthening your legs. You don't move the ground while you are runing outdoors. For this reason it is good to go for a motorized treading machine than a non-motorized one if you are not in a good shape already.