Induction week - the first week of uni - is a time to meet your fellow course mates, learn about what you will be studying, get to know campus, and the area you are studying better, and have a bit of fun before the hard work starts.
Academically, it depends on your course - for me, my course is funded by the government and I am accountable for all days - I have to sign a register to show attendance for each session or they withdraw my funding. I also have lectures etc that are compulsory and are within 'freshers week' - so when everyone else is out getting free pens and pizza from local shop stalls in the Fresher's fair, I'm studying. It's also the best time to get hold of the books / equipment for your course, sort out how the library works, set up study groups / study buddies, etc.
Durham is an academic uni, and they would probably not take kindly to you skipping a week because you had (correct me if i I'm wrong) booked the wrong week to attend the university. It is your responsibility as an adult - as you are 18+ going to uni - to be able to get yourself to the uni in a timely fashion and it will reflect badly upon you by doing this on the first week. In the short term, it sucks to lose some / a lot of money by transferring the flight, however in the long run you will come across as a more responsible student and more mature adult if you deal with your mistake and contact the airline provider and fix the error you made in booking the wrong week. Every university has the list of term dates on their websites, and it costs a lot of money to go to university, so they will be unsympathetic to you not checking the details. Also if you don't show up without contacting the university they will likely CANCEL YOUR PLACE. If you really cannot afford the transfer fee, speak to the university, explain your error and intended attendance date to take ownership of your mistake.
Unfortunately you need to pay the extra, cut your losses and check before you book next time using it as a learning experience. Hopefully this has helped you understand the importance of the first week, particularly if you were lucky enough to be given a place at such a reputable university. Also culturally in the UK - and places like Durham, Cambridge, Oxford University in particular - they really don't like people being late as it's so disrespectful and rude to do so.