Is it smart to watch a 4k video on a 1080p monitor versus watching it in 1080p (especially on youtube)? Is there any difference for the human eye?
No, there is no difference to the human eye (or exceptionally minimal). Your monitor can't display higher quality than it is capable of displaying. The only advantage you would have is if you were to zoom in on part of the image, you would have more detail when you zoomed in.
The exception to this is if the 4k stream uses enough extra bandwidth to reduce the number of artifacts present in the stream, but for a well configured stream this shouldn't be a significant factor. For Youtube settings, that may or may not be noticeable depending on your sensitivity to artifacts and the type of content you are viewing.
There's not a lot of point to it. You're using more network bandwidth to download it, and your computer is working harder to display it, but no, you get no additional visible quality from it. In fact, it's likely that you'll get lower quality because the computer has to downsample it before it can display it.
Others have said the answer is simply no, but that's not entirely true. You can get some slight improvement from downscaling; errors and artifacts in the stream will be less visible, but this slight improvement usually isn't worth the increased bandwidth usage and CPU usage. Effective downscaling also requires graphics which can manage it, Nvidia GTX GPUs can do this, I'm not sure about AMD or Intel graphics.
On youtube? It depends.
Youtube recompresses video uploads to reduce space and store a version of the video which requires the least CPU to decompress. I've uploaded 1080p videos which later look horrible in the youtube version. And the same happens with 4k video.
Having said that, it is very unlikely youtube will increase the quality of the video streams, which means 1080p video will always look as horrible as it looks now. So in a way, watching a 4k poorly compressed video stream on a 1080p monitor may look more pleasant if the video doesn't have much movement. When the video is still, a 4k low bandwidth video will be able to provide a more detailed picture which scaled down will look better than a low bandwidth 1080p video. But if you are looking at action footage, 4k youtube is only going to bring you higher visual artefacts.
A good example of how horrible youtube recompression is can be seen atwhich is a 4k video comparison of the Samsung Galaxy Alpha and Sony Xperia Z3 Compact. Watch the 4k youtube video and then download the raw 4k footage from the provided mediafire links. The most noticeable artefacts are in the lower part of the screen representing the Sony Xperia version: the 4k youtube version shows terrible compression artefacts around the shadow of the cameraman, but the raw 4k version is smooth as silk and the soft shadows look gorgeous.
If that doesn't convince you, think of it this way: the download size for the 4k youtube version is around 600MiB, the download size for the raw 4k video is 1.5GiB. So essentially anything you watch on youtube is one third the quality you get from the camera.
As with the general question of watching 4k on a 1080p monitor, yes, there is difference, there are comparisons of video between prosumer 1080p cameras and mobile 4k cameras and the 4k mobile version downscaled to 1080 looks better because it has sharper edges. But of course the difference is more notable on a 4k screen, and as other say, really good 1080p cameras are still much better than crappy 4k ones.
These answers are all very misleading, you'd think after 5 years someone would've told the truth, or did some research on this. Yes running your youtube videos over your monitors resolution WILL increase the clarity of the video. This is because youtube videos have a very low bit rate to allow for the platform to stream as many videos on an hourly basis that it does. As resolution increases for youtube playback, so does the bitrate, and so does definition, or clarity of the video. Im sure everyone has paused a 1080p youtube video when something is moving, and noticed a very harshly pixelated motion blur effect. This is greatly reduced at 1440p and 4k resolutions respectively, because more pixels are being pushed per second, being the bitrate.