I own a few games which I can't seem to load or run from the disk. Is it possible that a same game, purchased from GOG, will run even if my boxed version won't?
What games? How old?
If they're DOS games, or early Windows games that contain 16-bit code, they can't be run on 64bit Windows natively and have to be run inside an emulator like DOSBox. GOG usually puts the DOS games they sell inside DOSBox, though they occasionally use ScummVM instead. They try to make it look like the game is running natively in Windows, but it really isn't.
If you're talking about games made for Windows 98 or XP, GOG may have incorporated patches or tweaks that get rid of some of the problems with running the games on a modern computer. Sometimes it is the copy "protection" that prevents a game from running on a new computer, and GOG supplies their games without any copy "protection" that can block you from playing your legally purchased games.