I'm gonna break this stuff down for you guys and its a headache if you don't read through everything.
First off
P = PROGRESSIVE
I = INTERLACED
-What the hell is progressive and interlaced?
Progressive is though of as a still shot. It is a full picture, 1 frame. That is the most basic way to put it.
Interlaced video contains 2 fields. An odd field and an even field. In a 480i video, every other line is positive or negative. This little drawing I snatched up from google will give you a visual idea
As you see, interlaced video is almost like seeing 2 things at once, while progressive you are seeing full frames. When you deinterlaced interlaced footage, it pretty much becomes 30P.
So you ask why might you shoot in 60i over 30p? 60i is smoother and there is also more data there so if you decide to slow mo a clip it will be way smooth than the 30p clip.
...I'm going to get back to this because I'm going to eat dinner, haha.