If you’re lucky, the video might actually continue from where you stopped, otherwise you’d have to scrub through the video to get to that point. However, it seems that YouTube has introduced a new “Continue Watching” feature for its web-based player. Now whenever you stop watching a video on iOS or Android and launch YouTube.com on your computer, there will be a mini-player displayed in the bottom right corner of the screen.
This is essentially the video you were just watching that is now available for you to continue watching on the web and on your computer. Obviously you’d have to be signed into your Google account on your phone and web for this to work. According to 9to5Google, this feature also works for YouTube on Android TV.
However, it doesn’t seem to work in the opposite way, meaning that if you want to continue videos you watched on the web on your phone, then you’ll have to go about it using the old method which involves going into your Watch History to look for it.
YouTube is in the process of rolling out these changes to users, so if you don’t see it yet don’t worry as it should eventually find its way to you.