Netbooks are low-end and every way inferior to notebooks, but comes in a much smaller form factor, lighter and easier to carry. They run on mostly AMD chipsets to achieve affortability but suffers on performance. Although now with AMD's APU A-series chipset they perform quite well with graphics and fully capable with non-demanding games with graphics turned down. (I don't know how AMD APU will do with minecraft though, that game is a bitch to run)
Ultrabooks are high-end expensive thin laptops with mostly like intel chipsets and SSD storage, without dedicated graphics they might not even match AMD's APU performance in netbooks, although intel i-series chipset do perform much better with everyday task than AMDs. In order to build this super fast and thin shape they use SSDs and most like only have somewhere from 128~256Gb storage before getting ridiculously expensive. I can't even store movies in them, let alone install games.
Both of these will never be on part with notebooks, but they are not trying to be because they are aiming for convinence rather than functionality, so don't expect too much, especially in the gaming department, though low-end is passible. If you guys still live at home and has a desktop/more powerful laptop for other stuff either netbooks and ultrabooks are perfectly fine.
Although one of the biggest problem for both is battery life. Netbooks often require extended batterys and ultrabook with higher-end models which adds to their cost. Some of the cheaper ones don't even break the 4 hour mark, making them for long hours at school undesirable. Having to carry the powerbrick to school with me kind of defeating the purpose of having a lighter laptop :V.