Shooting in 60p
Over the past several years shooting corporate video (yippee!) I noticed that I use more and more 60p, or 60 frames per second (fps). I always use 30p (30fps) in normal shooting like a talking head interview or even for b-roll when the camera isn't moving much. But anytime I shoot something and the camera is moving and I need to see detail - for instance, a close-up pan of a product - I always now use 60p.
For those who are just starting out in video, when you shoot 60p you're shooting twice the amount of frames per second, so you're capturing double the information. That's why on a lot of modern DSLRs like a Nikon D800, you can only shoot 60p in the 1280x720 HD frame size. I believe newer DSLRs like the D810 can shoot 60p at full 1920x1080 HD.
60p is important if you want to capture motion without the blur associated with motion captured in 30p or 24p. You can't capture the same quality with either of those frame rates. You also cannot capture the same type of footage with 60p that you can with 30p or 24p, so you're forced to switch often if you only have one camera.
Some advantages of shooting 60p
60p captures more detail of moving things and of still things when the camera is moving, allowing the viewer to see more detail and enjoy the shot that you worked so hard to create.
60p allows you to slow down motion to create neat slow motion effects. If you want really cool slow motion like you see in films, you need even more frames captured, like 120p. You can only do this with more expensive professional cameras.
60p allows you to correct rough handheld shots better than lower frame rates. The stabilization feature in FCPX works wonders with 60p footage, whereas with lower frame rates it kinda stinks.
Ever tried using frame blending to smooth over 30p or 24p footage? It usually looks like crap but maybe just slightly better than what you had before. 60p footage helps you use frame blending less.
Some disadvantages of shooting 60p
You capture more information with 60p, so many cameras can only handle shooting it in 720p, not full 1080p. This sucks because you're usually shooting in 1080p so you'll have to convert your 720p footage to 1080p which will hurt the quality a bit. And with the rate of technology these days, 720p is becoming more and more a neglected format. Most people want at least 1080p if not 4K. But for online videos 720p is plenty.
The lowest shutter speed while shooting 60p is 1/60th of a second. This is because if you shoot any lower of a shutter speed you'll induce motion blur and camera companies don't won't amateurs thinking their cameras are broken. In more professional cameras you can shoot at lower shutter speeds with motion blur - this is even a technique used a lot in films - but if you want that you'll have to get something besides a consumer DSLR for video, or get the magic lantern hack for your 5D Mark ii or iii.
With that higher shutter speed you lose some low light ability of your DSLR and you'll have to turn up the ISO or open up your aperture, but general it's not much of a disadvantage unless you're shooting in super low light.
An easy way to see what 60p and higher frame rates can do is whip out your iPhone 5s or 6 and go to the camera app. You'll see that you're able to shoot slow mo video. The iPhone actually uses 120p to make slow motion video and it does a pretty good job at it, especially in good light. For outdoor slow motion shots you'll get some pretty good results, even stuff you can intermix with DSLR footage. If you need shallow depth of field you'll have to get a real camera for that.