If using legacy BIOS the MBR partitioning is required. If using UEFI BIOS then GPT partitioning is required. Since MBR booting can also be used with GPT partitioning it seems simplest to use GPT to avoid problems as tech continues to move forward.

For a GPT partitioned disk, the detailed difference between using UEFI and legacy BIOS with MBR boot is that when creating the GPT partition table, if you tell it to create a protected MBR sector then the installer will normally use UEFI. If the MBR sector is not protected then most will install in MBR mode.

Another difference that may affect you is that the installer on some distros, when installing in auto-partitioning mode, may create /boot and / as the only partitions they default to. This means that /home is not a separate partition and can become in issue if there is a need to reinstall. Some create /home as a separate partition.

As beachboy2 said, mint creates only the 2 partitions so if you want /home separate you will need to manually create the partitions.

the second partition, efi esp, is subdirectory under the main /boot partition?

It can be, and I prefer to do that, or the /boot partition can be made the efi partition. If the /boot is made efi then everything there is in a vfat partition.

what then is stored in /boot if its not a boot partition?

If separate partition from /boot/efi then /boot can be ext4 and stores the kernel boot images. Booting is actually 2 stage. First is the grub boot loader which then loads the kernel image.

/boot/efi must be vfat, flagged as esp, and contains the grub boot loader. The grub boot loader for efi must be in a fat32,16,12 (now commonly vfat AKA fat32) partition that the uefi bios is able to access. The grub boot loader is able to do more than the bios for access so it can access the kernel boot images on either vfat or ext4 but cannot access an LVM partition so it is your choice to use /boot and /boot/efi or only /boot when you do the install.

The difference as I mentioned is that if you use only /boot then the entire partition has to be vfat and flagged as esp. If using /boot/efi as a separate partition then only that smaller partition is required to be vfat and flagged as esp.

Regardless of what you choose for partitioning the directory structure of /boot/efi (and subdirectories) will be created for the grub boot loader, and the kernel images will be in /boot.

/boot is getting less important than it once was, because of the move to UEFI booting. Originally this directory contained everything you needed for booting your system: the kernel, the initrd image, kernel configuration and system map files, and files used by the bootloader (either LILO or GRUB). But with a UEFI boot, the kernel and initrd go onto the EFI system partition along with a either elilo or GRUB as bootloader. There's not much left for the old /boot directory to do except host the GRUB modules and provide a mountpoint for the esp when required.

Different distros treat this differently. AIUI most of the ubuntu style systems use only /boot. Most of the fedora/centos style systems use /boot and /boot/efi. This easily allows installing dual boot with windows and using the already small (200 to 250MB) efi partition as /boot/efi. All the kernel stuff you mentioned are in /boot. I don't see that as making /boot less important. Even the uefi bios is still not able to actually load the kernel image and needs the intermediary boot loader.