This is somewhat to be expected. The resolution of the monitor has likely increased by quite a bit, whereas the resolution of the videos has not. So you will tend to notice graphical defects more because of a sort of optical illusion.
Our brain uses things around us to determine focus. One of the things a lot of people seem to first remark when they get glasses for the first time is along the lines of, "The trees have leaves!" If I am trying to focus on something, internally my brain is also processing the peripheral vision from my eyes to help determine whether or not something is in focus. If everything in the periphery is sharp, but what I'm looking at is blurry, it'll give you a headache, and the other way around. It's also related to sea sickness, but that's a whole other topic. If the rest of your monitor is quite sharp relative to the video you're playing, that will be noticed by you more so than on a monitor where the overall image quality was much poorer.
Some of the rest might have more to do with some of the tricks companies use so they can advertise those ridiculously low response rates on monitors than your video card. Check your monitor's settings, and see if there's some kind of pixel overdrive setting or something of that sort. If so, try turning it off and see if does anything for the video quality (good or bad). If it does, the issue is with your monitor, and you probably buying a cheap low quality one. You might be able to ease the symptoms, but you'll likely never get rid of them. In this situation, either learn to live with it or take the monitor back and get a better quality one.
If it has no effect at all, you can try upgrading your video drivers. It may or may not help. You can also try the new high quality option on YouTube. Not sure about Netflix.
My guess though, is you're just more sensitive to this, and these issues were actually always there. Unless you're trying to watch these videos at a higher quality setting now that you have a new monitor, and then neglected to mention that little nugget.