I could go into specific examples but that would spoil the fun of seeing the movies for yourselves. It would be pointless to go into specifics due to plot summaries that can be found online.
Pseudo-prequels – God of Gamblers: The Early Stage and The Thing.
Suspicious sequels – An American Werewolf in Paris, The Rage: Carrie 2, Blues Brothers 2000, Basic Instinct 2, Wall Street: Money Never Sleeps, Predators, The Bourne Legacy, Mad Max: Fury Road, Terminator: Genisys, Independence Day: Resurgence, Flatliners and Star Wars: The Force Awakens.
The irony is that there are sequels which can’t be described as remakes because of how much time has gone by since the first movie. They are simply called rehashes (if not potboilers). After so many sequels being described this way, film-makers realized that the best way to avoid the stigma of a remake is to use the same characters or refer to the original in a way which suggests a direct continuation. As to why they do it like this, they understand that people despise the notion of actors being replaced.
This happened with the Star Trek reboot. I suspect that the latest Star Wars series is pretty much that (something that’s bound to get lampooned on South Park). This happens on television (e.g. the Hawaii Five-O reboot). The Simpsons was pretty much rebooted after the eleventh season (the true end which even the adaptation film-makers had referenced). Anyway, the film-makers hope that there are enough differences to the extent that people won’t describe their efforts as retreads.
Back to TV, there will eventually be a trend of cancelled TV serials having sequels in the form of delayed seasons (if not outright remakes). This is a method designed to justify refurbishing a TV series for the Blu-ray medium. This already happened with 90210. Some will argue that a spin-off is a sequel, but that’s like saying Elektra is a sequel to Daredevil. The ending of Twin Peaks hinted a follow-up decades down the line.
The best way to go about resurrecting a series is to explore crossover potential that never happened (even if it only happens as a TV movie). As far as reliving glory days, TV prequels are a surefire way to go depending on the subject matter. Bates Motel is the least vilified of them. Gotham is a classic series because it’s an origin story that kind of explains why something like Hannibal gets cancelled after three seasons in.