You should realize, the comments above are referring to 1080 60p. That's 1080 lines at 60 frames per second. 1080 24p and 1080 30p are part of the ATSC standard, but I have not heard of any use of these formats. 1080 60p is not part of the ATSC standard. It would take up twice the bandwidth of 1080i, and until better compression comes along, would generally not look good crammed into 6 MHz of bandwidth. MPEG-4 (not part of the ATSC standard) would be the minimum to do this, and preferably something better will become available. I doubt that we will see 1080 60p added to the ATSC standard in our lifetimes. OTA standards are not easily changed. OTA broadcasts may dwindle away anyway, and make the point moot. I doubt the cablecos or satcos will have the bandwidth to support this anytime in the next decade or two- if ever.
Fios- no problem.
When you see "1080p" in an HDTV ad, they are referring to the DISPLAY, not the signal. Some will accept a 1080 progressive signal at various frame rates. I don't know of any cameras in general use that capture content at 1080 60p, so something provided at that rate is likely computer generated or processed to create that type of signal. Heck, cameras capturing 1080x1280 & 1080x1440 are still in common use.
In theory, 720p displays motion better than 1080i, and 1080i displays still scenes better than 720p. But the actual camera, processing, transmission and a myriad of other factors may overwhelm this theoretical distinction. And, of course, multicasting throws a wrench into the whole works, although 720p is reputed to tolerate multicasting better due to it's slightly lower bit rate requirements.
And the size of the display matters. At 19", I don't think most people would see much difference between 480, 720 and 1080 lines. 27" - not a whole lot of difference between 720 & 1080. Maybe the difference between 720 & 1080 doesn't become apparent until 42". It's certainly there at 61". Depends upon your eyes and the viewing conditions. Don't get too wrapped up around bigger numbers. You could pay for more than you can see if you're buying a small TV.