When watching a video on YouTube in 1080p 60 FPS the video runs very smooth and looks nice. When I play games on my PC with more than 60 FPS it does not feel as nice.
Is there a difference between YouTube's 60fps and 'actual' 60fps?
If you're comparing PC games to Youtube video shot with a camera (as opposed to Youtube video of games captured via HDMI), then you're probably noticing the natural motion blur that cameras introduce. Gaming hardware can't reproduce this effect very well because it either has to render intermediate frames and blur them together or perform "optical flow" or vector-based analysis in real time.
If you are, however, asking about Youtube videos of video games (which presently comprise most of Youtube's 60p content), then my only suggestion is perhaps a performance difference between your machine and the ones in the videos.
My best guess is that your computer may be having trouble actually producing 60fps fast enough. Try turning on triple buffering and see if that helps. If the buffer doesn't fully fill before a frame goes to display, you will get an artifact known as screen tearing (some of the buffer has the current frame, some has the previous). Triple buffering fixes this by fully pre-rendering a frame in advance but results in increased latency due to the extra processing delay.
Additionally, if you are comparing videos from games that other people are running vs your own recordings, the game quality settings make a huge difference. Things like Anti-aliasing are very demanding on computer hardware, but make edges look far smoother. Motion blur and shadows are also rather processing intensive effects that make a big difference. If your system isn't super, super powerful, keeping up with 60fps while also keeping those settings on all the way is quite hard.
It's very likely that your framerate drops below the refresh rate of your display while you have v-sync on which can produce a noticeable stuttering or input lag in that situation.
Disabling vsync will probably give you a more "fluid" experience but can introduce screen tearing, which vsync is meant to prevent by locking the framerate to the refresh rate of your monitor, though if your pc doesnt render fast enough it waits for a short amount of time which you can notice in this lag/stuttering appearing.
Unlike games, the framerate of a youtube video in this case is consistent at 60fps and isn't depending as much (by far) on the rendering speed of your computer as a game does.
Do you play your game with v-sync on? If no, your frames don't line up with when the video card actually sends an update to the monitor, so you get tearing. This can and does make things look less smooth.
Maybe the videos you're watching were made by people that had vsync on at 60Hz, to avoid any tearing in any of the captured frames?
Check out this article on g-sync / freesync, aka variable sync / variable frame-rate monitors. The author does a good job explaining v-sync, input lag, and all that. With diagrams even.
(And as an update since that article came out, AMD now does have their own version of this. And variable-refresh-rate is optional in the VESA DisplayPort 1.2a spec, so it's on the road to becoming an industry-standard feature.)
Adaptive sync gives you the same (or even better) smoothness you'd get with a game that stayed pegged at 60Hz with vsync on. But it's still smooth with settings pushed up so high that you get occasional dips below 60FPS, so you can crank the settings up and have your game look awesome, while still being smooth with minimal lag between keyboard/mouse input and screen output.
60 fps videos on YouTube are progressive while many video games cheat with interlacing.
Another potential cause is the refresh rate of your monitor. Most monitors refresh at 60Hz, meaning 60 times a second. But because of the way GPUs cycle when rendering video games, you need a 120Hz monitor to appreciate 60 fps.