I've mentioned this before in passing ( cf , also ) but today I feel like saying it explicitly: our habit with all of our machines is to never apply a kernel update without immediately rebooting the machine into the new kernel. On our Ubuntu machines this is done by holding the relevant kernel packages; on my Fedora desktops I normally run 'dnf update --exclude "kernel*"' unless I'm willing to reboot on the spot.
The obvious reason for this is that we want to switch to the new kernel under controlled, attended conditions when we'll be able to take immediate action if something is wrong, rather than possibly have the new kernel activate at some random time without us present and paying attention if there's a power failure, a kernel panic, or whatever. This is especially acute on my desktops, where I use ZFS by building my own OpenZFS packages and kernel modules. If something goes wrong and the kernel modules don't load or don't work right, an unattended reboot can leave my desktops completely unusable and off the network until I can get to them. I'd rather avoid that if possible ( sometimes it isn't ).
(In general I prefer to reboot my Fedora machines with me present because weird things happen from time to time and sometimes I make mistakes , also .)
The less obvious reason is that when you reboot a machine right after applying a kernel update, it's clear in your mind that the machine has switched to a new kernel. If there are system problems in the days immediately afterward the update, you're relatively likely to remember this and at least consider the possibility that the new kernel is involved. If you apply a kernel update, walk away without rebooting, and the machine reboots a week and a half later for some unrelated reason, you may not remember that one of the things the reboot did was switch to a new kernel.
(Kernels aren't the only thing that this can happen with, since not all system updates and changes take effect immediately when made or applied. Perhaps one should reboot after making them, too.)
I'm assuming here that your Linux distribution's package management system is sensible, so there's no risk of losing old kernels (especially the one you're currently running) merely because you installed some new ones but didn't reboot into them. This is how Debian and Ubuntu behave (if you don't 'apt autoremove' kernels), but not quite how Fedora's dnf does it (as far as I know). Fedora dnf keeps the N most recent kernels around and probably doesn't let you remove the currently running kernel even if it's more than N kernels old, but I don't believe it tracks whether or not you've rebooted into those N kernels and stretches the N out if you haven't (or removes more recent installed kernels that you've never rebooted into, instead of older kernels that you did use at one point).
PS: Of course if kernel updates were perfect this wouldn't matter. However this isn't something you can assume for the Linux kernel (especially as patched by your distribution), as we've sometimes seen . Although big issues like that are relatively uncommon.