Video help? recording in 4k?
What you see on the larger screen is the real image quality (to the limits of that screen's ability to show it). For better results, use a 4k camcorder instead of using a phone.
It is all a matter of screen size.
Your phone's screen is so small that all the details that made you look "rough" on the relatively large monitor in your laptop. This is why so many people think that the $8 camera module in their $1,000 smartphone takes such good photos. If they were to ever see those "great" shots printed 8x10 or shown on a decent 17"~27" monitor, they wouldn't feel the same way about their phones.
I agree with Al. If you are playing the raw video files on your laptop some video players can look rough. Sometimes the laptop itself will struggle to play a raw 4k video in any player. The VLC player is the best out there so start with that. Otherwise another good test is upload your raw video to youtube and then play it on your laptop and see how it looks. A lot of vlogers record their videos with a lot of special camera settings and do post processing filters setup in their editing software specifically for the camera so you will need to research that too.
The software you're using on your laptop possibly isn't capable of playing 4K content. You could try using something like VLC Media Player (https://www.videolan.org/vlc/index.html). Even if your phone isn't playing 4K, it'll look better since the screen is smaller and the image appears to be sharper, even if it isn't.
4k is 3840 or 4096 vs 2160 resolution, way more than your laptop and way more than your phone so, the difference will be in the resolution and in any codecs or compression that are used. Your phone screen is, what, 10% of the area of your laptop screen? You lose lot of finer detail that way