The apartment we're staying in now is furnished, which is a great luxury. It includes everything from silverware, dishes and cups to towels and sheet.
However, from our stay in Somerset last year we learned that even in a fully furnished temporary apartment there are still things that are worthwhile bringing in. Most important is the big box with spices and herbs and a good kitchen knife (the knifes that come with the apartment are not very good). We also brought our favorite coffee mugs.
When we left Durham, now almost 6 weeks ago, we prepared a few boxes with things, mainly cooking equipment. Unlike last time, we did not have to get a new coffee machine.