According to man, "update-grub" is merely a stub for "grub-mkconfig ....". There should be no big difference in execution time between two commands. It is confirmed by direct tests below.
dt is an alias, which displays current date and time (timestamp).
update-grub
┌──[just]@[alexaxc]:~$
└─> dt && sudo update-grub && dt
Current datetime:
2021-10-09 19:38:41
Generating grub configuration file ...
Found theme: /usr/share/grub/themes/artix/theme.txt
Found linux image: /boot/vmlinuz-linux
Found initrd image: /boot/intel-ucode.img /boot/amd-ucode.img /boot/initramfs-linux.img
Found fallback initrd image(s) in /boot: intel-ucode.img amd-ucode.img initramfs-linux-fallback.img
Warning: os-prober will be executed to detect other bootable partitions.
Its output will be used to detect bootable binaries on them and create new boot entries.
Found Devuan Plasma (testing) on /dev/sda1
Found Devuan Cinnamon (testing) on /dev/sda10
Found Devuan Gnome (testing) on /dev/sda11
Found Debian Plasma bookworm/sid on /dev/sda12
Found Debian Cinnamon bookworm/sid on /dev/sda13
Found Debian Gnome bookworm/sid on /dev/sda14
Found Debian Mate bookworm/sid on /dev/sda15
Found Void Plasma on /dev/sda2
Found Void Cinnamon on /dev/sda5
Found Void Gnome on /dev/sda6
Found Void Mate on /dev/sda7
Found Artix Plasma (rolling) on /dev/sda8
Found memtest86+ image: /boot/memtest86+/memtest.bin
done
Current datetime:
2021-10-09 19:49:26
update-grub took roughly 19:49 - 19:38 = 11 minutes to complete.
grub-mkconfig
┌──[just]@[alexaxc]:~$
└─> dt && sudo grub-mkconfig -o /boot/grub/grub.cfg && dt
Current datetime:
2021-10-09 19:54:04
Generating grub configuration file ...
Found theme: /usr/share/grub/themes/artix/theme.txt
Found linux image: /boot/vmlinuz-linux
Found initrd image: /boot/intel-ucode.img /boot/amd-ucode.img /boot/initramfs-linux.img
Found fallback initrd image(s) in /boot: intel-ucode.img amd-ucode.img initramfs-linux-fallback.img
Warning: os-prober will be executed to detect other bootable partitions.
Its output will be used to detect bootable binaries on them and create new boot entries.
Found Devuan Plasma (testing) on /dev/sda1
Found Devuan Cinnamon (testing) on /dev/sda10
Found Devuan Gnome (testing) on /dev/sda11
Found Debian Plasma bookworm/sid on /dev/sda12
Found Debian Cinnamon bookworm/sid on /dev/sda13
Found Debian Gnome bookworm/sid on /dev/sda14
Found Debian Mate bookworm/sid on /dev/sda15
Found Void Plasma on /dev/sda2
Found Void Cinnamon on /dev/sda5
Found Void Gnome on /dev/sda6
Found Void Mate on /dev/sda7
Found Artix Plasma (rolling) on /dev/sda8
Found memtest86+ image: /boot/memtest86+/memtest.bin
done
Current datetime:
2021-10-09 20:05:02
grub-mkconfig took roughly 20:05 - 19:54 = 11 minutes to complete.
There's no difference in timings between two commands in my environment. Both take 11 minutes to complete here.