While 1080i has 1080 lines of resolution, 720p has only 720 lines. Theoretically, more lines means higher resolution which means better picture quality. However, on the whole, it is practically impossible for an average customer to tell the difference between the picture quality of 720p and 1080i.
Contents: 1080i vs 720p
edit 1080i vs 720p - Differences in display Technology
edit Interlaced vs. Progressive display technology
1080i - Interlaced
1080i represents 1,080 lines of resolution scanned in alternate fields consisting of 540 lines each. 1080i is the most commonly used HDTV format, and has been adopted by most television broadcast, cable, and satellite outlets as their HDTV broadcast standard.
720p - Progressive
720p represents 720 lines of resolution scanned sequentially. In other words, all lines are scanned in progressively, providing a more detailed high definition video image compared to interlaced display technology of the same resolution. Progressive scanning reduces the need to prevent flicker by filtering out fine details, so spatial (sharpness) resolution is much closer to 1080i than the number of scan lines would suggest.
edit Video explaining the difference between 720p and 1080i
Here's a video that explains the differences between 1080i and 720p.