Aditya_26 :
Sir/ Ma'am,
I can discernibly tell the difference between HD and FHD in general, so as an average laptop user I am sure I would be able to tell the difference while watching things. However, if as you say on a laptop it shouldn't matter too much, I suppose I may be able to compromise
Are you referring to watching a HD movie / video on a FHD monitor or HDTV?
In that case, most people can easily tell the difference because when you watch a 720p movie / video on a 1080p screen video quality takes a steep dive because the smaller video resolution must be stretched to fill the larger screen resolution. This is done through a process called interpolation which attempts to estimate where pixels are place on the screen to show a larger image. This causes video to be less defined and fuzzy.
Generally speaking, the higher the resolution something is the more detailed it is. Stretching a small video resolution to fit a larger screen resolution does not instantly create detail in the video image itself. It's like taking a 9x6 hand knit blanket and stretching it to 18x12. The more you stretch that blanket the larger the space in between the yarn strands will be.
However, when taking a higher resolution video and displaying it on a smaller resolution screen the video will not look as bad as the above case. There is video interpretation involved, but instead of making the video larger by filling in gaps when the video is stretched, details (the pixels themselves) are removed so that the 1080p video can be shrunk down to fit on a 720p or 768p resolution screen. Sure you will loose detail, but the video will still look much sharper compared to stretching a video.
The above is the reason why DVD movies (720p) look terrible when you attempt to watch them on a 1080p screen. They will look even worse if you attempt to watch a DVD movie on 4k HDTVs / monitors.
Between the choice of buying a laptop with a 768p or 1080p screen, I would always choose a laptop with the 1080p screen because when I work on spreadsheets or word documents the higher resolution screen will allow me to see more information on the screen at one time. That means less scrolling back and forth when working with spreadsheet that uses many columns and/or rows and large word documents.
One of my laptops (from 2011) only has a 1366x768 screen and I am definitely going to be replacing it. The questions that remain are how much I am willing to spend on another laptop and if I should wait for laptops coming out later this year with Intel's 7th generation Kaby Lake CPUs.