Its resolution v smooth movement (temporal information).
60i is 60 samples taken per second, but each sample is only half the resolution of the full screen. This gives you "video" motion which is smooth, even with fast camera pans or fast moving action. Because each sample is only half the resolution, when something moves it is only captured with half the detail. You can see this sometimes at the start of a slow camera pan resolution can drop, when you stop panning and hold the camera steady the picture resolution improves. This wasn't so noticeable with 60i at SD resolutions but more noticeable with HD. Also 60i is optimised for interlaced displays, but we don't use them anymore, we have progressive displays. This means the TV has to convert the 60i to 60p using various tricks, which can degrade the picture quality or introduce artefacts.
30p is 30 samples taken per second at full resolution. Because of the lower sample rate, movement and camera pans may not look so smooth, so you need to be careful with 30p not to have two much fast motion or fast camera pans, so not the best format for sports for example. 30p displays better on modern TVs as it is progressive to progressive. When something moves the resolution doesn't drop, but ironically our eyes, especially with LCD TVs, trying to track the motion with a low sample rate of 30 frames a second, causes us to perceive a blurring anyway. (We are moving our eyes over a series of static pictures, so it's like camera blur).
30p is great for the Internet and YouTube however as it is a good compromise, and internationally the majority of monitors refresh at 60p, so to display 30p on 60p monitors each frame is shown twice.
The best option is 60p if you have it, this is the best of both worlds, although not good for the internet due to the bandwidth requirements and can be difficult to edit and maintain 60p playback on your TV.