I am not sure about exactly differences of those two applications you have mentioned but in most cases difference in app names are either in possibility that one application record stream directly using original video link Vs the other which can not do that so you need to record it by recording what is on the screen. Or, in the other case, that one application is capable only to record video stream while other can do some other things like converting recorded files and has some other tools more than the first one.
Answer on your second question would be that you will have to try every website you want to get. Speaking from my experience, most of the live streaming videos you will not be able to record if you leave the page with video or stop watching. If we speak about videos that are not live than most likely you will able to get them without having to stay and watch.
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