The difference between 480i and 480p (assuming that the source material was at this rate) is that the interlaced is shown at 60 half frames (each is called a field) per second, and each of these half frames are shot 1/60 of a second apart. So when you combine 480i to be shown on 480p equipment the two half frames get put together, but as they were not shot at the same time, they are slightly out of sync - especially for fast motion shots. The result is that the image looks slightly fuzzy, as the two half frames don't line up precisely.
This is because material originally shot at 480i was recorded at 60 fields per second. The video being put to 1080i or 1080p is not, it's either a film transfer (24fps) or shot with an HD camera. If that HD camera is a 720p, then your only scaling the data, not de-interlacing it. The only chance would be if it's shot with a 1080i cam, but as I understand it they save the data as 60 fields per second, but they're not shot with a 1/60th second offset between fields, so they combine better.