University halls and a private room are different contracts, not two versions of the same thing.
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Halls are run by the university. First-years are usually first in line, and the room may be assigned to you. You will meet a lot of people on the same corridor in the same position. Some halls include bills. Some do not. Ask.
A private room here is one you choose, in a building you can look up on a map. The weekly price includes gas, electricity, water, Wi-Fi and contents insurance. You can apply in any year of the course. The building has a gym, a study room and staff on site overnight.
Pick halls if you want the university to place you with other first-years. Pick a private room if you want a known address and a rent figure that will not grow a utility bill later.