A shutdown is simply that a power down. Does not make you feel refreshed and ready to go - you just shut down.
Basically, when you restart your computer, you remove the random, unimportant, and temporary data that may be causing issues with your device. Any bad code that is causing issues will be removed When you restart, the computer or device will bring the system up from scratch, restarting all the software so it will work just as well as it was working before.
So, while there is no answer to "how often should I restart my computer?" - I tend to restart my computer every other week.
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