Thanks for trying Peppermint and welcome to the Forums.
I believe you can do either. If I key any errors, I am sure I'll be corrected forthwith. mintUpdate will be a bit more cautious and automatic, not grabbing every single update. apt update && apt upgrade should pull in the what mintUpdate does plus levels 4 and 5. Some of our users have elected to disable mintupdate and just use apt or synaptic. It is your choice. There are existing topics available for preventing mintUpdate from running. I would recommend just letting mintUpdate do its thing.
As Kendall mentions in another topic,
"...mintUpdate breaks down updates according to the likelihood that they may cause lower level system problems into 5 categories and only installs down to level 3 as a degree of insurance over your system's stability. 99.9% of the time it's a little over cautious, but in some cases the extra layer of safety is either needed or desired."